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

%I A078937 #17 Mar 28 2024 11:58:57
%S A078937 1,2,1,6,4,1,22,18,6,1,94,88,36,8,1,454,470,220,60,10,1,2430,2724,
%T A078937 1410,440,90,12,1,14214,17010,9534,3290,770,126,14,1,89918,113712,
%U A078937 68040,25424,6580,1232,168,16,1,610182,809262,511704,204120,57204,11844,1848,216,18,1
%N A078937 Square of lower triangular matrix of A056857 (successive equalities in set partitions of n).
%C A078937 First column gives A001861 (values of Bell polynomials); row sums gives A035009 (STIRLING transform of powers of 2);
%C A078937 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
%C A078937 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]
%F A078937 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
%F A078937 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.
%e A078937 [0] 1;
%e A078937 [1] 2, 1;
%e A078937 [2] 6, 4, 1;
%e A078937 [3] 22, 18, 6, 1;
%e A078937 [4] 94, 88, 36, 8, 1;
%e A078937 [5] 454, 470, 220, 60, 10, 1;
%e A078937 [6] 2430, 2724, 1410, 440, 90, 12, 1;
%e A078937 [7] 14214, 17010, 9534, 3290, 770, 126, 14, 1;
%e A078937 [8] 89918, 113712, 68040, 25424, 6580, 1232, 168, 16, 1;
%p A078937 # Computes triangle as a matrix M(dim, p).
%p A078937 # A023531 (p=0), A056857 (p=1), this sequence (p=2), A078938 (p=3), ...
%p A078937 with(LinearAlgebra): M := (n, p) -> local j,k; MatrixPower(subs(exp(1) = 1,
%p A078937 MatrixExponential(MatrixExponential(Matrix(n, n, [seq(seq(`if`(j = k + 1, j, 0),
%p A078937 k = 0..n-1), j = 0..n-1)])))), p): M(8, 2);  # _Peter Luschny_, Mar 28 2024
%o A078937 (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 */
%Y A078937 Cf. A056857, A001861, A035009.
%Y A078937 Cf. A078938, A078944, A078945, A000110.
%Y A078937 Cf. A078937, A078938, A129323, A129324, A129325, A027710.
%Y A078937 Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.
%K A078937 nonn,tabl
%O A078937 0,2
%A A078937 _Paul D. Hanna_, Dec 18 2002
%E A078937 Entry revised by _N. J. A. Sloane_, Apr 25 2007
%E A078937 a(38) corrected by _Georg Fischer_, Mar 28 2024