A078937 Square of lower triangular matrix of A056857 (successive equalities in set partitions of n).

1, 2, 1, 6, 4, 1, 22, 18, 6, 1, 94, 88, 36, 8, 1, 454, 470, 220, 60, 10, 1, 2430, 2724, 1410, 440, 90, 12, 1, 14214, 17010, 9534, 3290, 770, 126, 14, 1, 89918, 113712, 68040, 25424, 6580, 1232, 168, 16, 1, 610182, 809262, 511704, 204120, 57204, 11844, 1848, 216, 18, 1

Offset

0,2

Comments

First column gives A001861 (values of Bell polynomials); row sums gives A035009 (STIRLING transform of powers of 2);

Square of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Apr 08 2007. Base matrix in A011971 and in A056857, second power in this entry, third power in A078938, fourth power in A078939

Riordan array [exp(2*exp(x)-2),x], whose production matrix has e.g.f. exp(x*t)(t+2*exp(x)). [Paul Barry, Nov 26 2008]

Links

Table of n, a(n) for n=0..54.

Formula

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,column] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1] - Gottfried Helms, Apr 08 2007

Exponential function of 2*Pascal's triangle (taken as a lower triangular matrix) divided by e^2: [A078937] = (1/e^2)*exp(2*[A007318]) = [A056857]^2.

Example

[0] 1;
[1] 2, 1;
[2] 6, 4, 1;
[3] 22, 18, 6, 1;
[4] 94, 88, 36, 8, 1;
[5] 454, 470, 220, 60, 10, 1;
[6] 2430, 2724, 1410, 440, 90, 12, 1;
[7] 14214, 17010, 9534, 3290, 770, 126, 14, 1;
[8] 89918, 113712, 68040, 25424, 6580, 1232, 168, 16, 1;

Maple

# Computes triangle as a matrix M(dim, p).
# A023531 (p=0), A056857 (p=1), this sequence (p=2), A078938 (p=3), ...
with(LinearAlgebra): M := (n, p) -> local j, k; MatrixPower(subs(exp(1) = 1,
MatrixExponential(MatrixExponential(Matrix(n, n, [seq(seq(`if`(j = k + 1, j, 0),
k = 0..n-1), j = 0..n-1)])))), p): M(8, 2);  # Peter Luschny, Mar 28 2024

Prog

(PARI) k=9; m=matpascal(k)-matid(k+1); pe=matid(k+1)+sum(j=1, k, m^j/j!); A=pe^2; A /* Gottfried Helms, Apr 08 2007; amended by Georg Fischer Mar 28 2024 */

Crossrefs

Cf. A056857, A001861, A035009.

Cf. A078938, A078944, A078945, A000110.

Cf. A078937, A078938, A129323, A129324, A129325, A027710.

Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

Sequence in context: A117852 A080245 A080247 * A167560 A132159 A112356

Adjacent sequences: A078934 A078935 A078936 * A078938 A078939 A078940

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 18 2002

Extensions

Entry revised by N. J. A. Sloane, Apr 25 2007

a(38) corrected by Georg Fischer, Mar 28 2024

Status

approved