A095674 - OEIS

A095674

Triangle read by rows, formed from product of Pascal's triangle (A007318) and Aitken's (or Bell's) triangle (A011971).

1, 2, 2, 5, 7, 5, 15, 22, 25, 15, 52, 74, 97, 97, 52, 203, 277, 372, 449, 411, 203, 877, 1154, 1524, 1948, 2209, 1892, 877, 4140, 5294, 6816, 8734, 10718, 11570, 9402, 4140, 21147, 26441, 33255, 41954, 52357, 62107, 64404, 50127, 21147, 115975, 142416, 175669, 217583, 269520, 328715, 378909, 379924, 285151, 115975

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

These triangles are to be thought of as infinite lower-triangular matrices.

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

From G. C. Greubel, Jan 05 2026: (Start)

T(n, n-k) = T(n, k).

T(n, 1) = A024716(n) - 1.

Sum_{k=0..n} (-1)^k*T(n, k) = A367775(n) - A367775(n-1).

Sum_{j=0..k} (-1)^(j+k)*binomial(k, j)*T(n+j, k) = A000110(n+k+1); with example: let b(n) = T(n, 2), then b(n+2) - 2*b(n+1) + b(n) = A000110(n+3). (End)

EXAMPLE

Triangle begins:

2, 2;

5, 7, 5;

15, 22, 25, 15;

52, 74, 97, 97, 52;

203, 277, 372, 449, 411, 203;

877, 1154, 1524, 1948, 2209, 1892, 877;

4140, 5294, 6816, 8734, 10718, 11570, 9402, 4140;

21147, 26441, 33255, 41954, 52357, 62107, 64404, 50127, 21147;

MATHEMATICA

a[0, 0] = 1; a[n_, 0] := a[n - 1, n - 1]; a[n_, k_] := a[n, k] = If[k < n + 1, a[n, k - 1] + a[n - 1, k - 1], 0]; p[n_, r_] := If[r <= n + 1, Binomial[n, r], 0]; am = Table[ a[n, r], {n, 0, 9}, {r, 0, 9}]; pm = Table[p[n, r], {n, 0, 9}, {r, 0, 9}]; t = Flatten[pm.am]; Delete[ t, Position[t, 0]] (* Robert G. Wilson v, Jul 12 2004 *)

PROG

(Magma)

A007318:= func< n, k | k le n select Binomial(n, k) else 0 >;

A011971:= func< n, k | k le n select (&+[Binomial(k, j)*Bell(n-k+j): j in [0..k]]) else 0 >;

A095674:= func< n, k | (&+[A007318(n, j)*A011971(j, k): j in [0..n]]) >;

[A095674(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 06 2026

(SageMath)

def A007318(n, k): return binomial(n, k) if k<n+1 else 0

@CachedFunction

def A011971(n, k): return sum(binomial(k, j)*bell_number(n-k+j) for j in range(k+1)) if k<n+1 else 0

def A095674(n, k): return sum(A007318(n, j)*A011971(j, k) for j in range(n+1))

print(flatten([[A095674(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 06 2026

CROSSREFS

Cf. A000110, A007318, A011971, A024716, A095675, A367775.

Row sums give A005494.

First column is A000110.

Sequence in context: A021447 A136536 A023507 * A207981 A058123 A035586

Adjacent sequences: A095671 A095672 A095673 * A095675 A095676 A095677

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane, based on a suggestion from Gary W. Adamson, Jun 22 2004

EXTENSIONS

More terms from Robert G. Wilson v, Jul 13 2004

STATUS

approved