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A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).

%I A292915 #25 Jun 13 2024 01:47:31
%S A292915 1,1,1,1,2,3,1,3,6,13,1,4,11,26,75,1,5,18,51,150,541,1,6,27,94,299,
%T A292915 1082,4683,1,7,38,161,582,2163,9366,47293,1,8,51,258,1083,4294,18731,
%U A292915 94586,545835,1,9,66,391,1910,8345,37398,189171,1091670,7087261,1,10,83,566,3195,15666,74067,378214,2183339,14174522,102247563
%N A292915 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(2 - exp(x)).
%C A292915 A(n,k) is the k-th binomial transform of A000670 evaluated at n.
%H A292915 G. C. Greubel, <a href="/A292915/b292915.txt">Antidiagonals n = 0..50, flattened</a>
%H A292915 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A292915 E.g.f. of column k: exp(k*x)/(2 - exp(x)).
%F A292915 A(n,k) = 2^k*A000670(n) - Sum_{j=0..k-1} 2^j*(k-1-j)^n. - _Seiichi Manyama_, Dec 25 2023
%e A292915 E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 3)*x^2/2! + (k^3 + 3*k^2 + 9*k + 13)*x^3/3! +  (k^4 + 4*k^3 + 18*k^2 + 52*k + 75) x^4/4! + ...
%e A292915 Square array begins:
%e A292915     1,     1,     1,     1,     1,      1,  ...
%e A292915     1,     2,     3,     4,     5,      6,  ...
%e A292915     3,     6,    11,    18,    27,     38,  ...
%e A292915    13,    26,    51,    94,   161,    258,  ...
%e A292915    75,   150,   299,   582,  1083,   1910,  ...
%e A292915   541,  1082,  2163,  4294,  8345,  15666,  ...
%p A292915 A:= proc(n, k) option remember; k^n +add(
%p A292915        binomial(n, j)*A(j, k), j=0..n-1)
%p A292915     end:
%p A292915 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Sep 27 2017
%t A292915 Table[Function[k, n! SeriesCoefficient[Exp[k x]/(2 - Exp[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t A292915 Table[Function[k, HurwitzLerchPhi[1/2, -n, k]/2][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%o A292915 (PARI) a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
%o A292915 A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n); \\ _Seiichi Manyama_, Dec 25 2023
%o A292915 (Magma)
%o A292915 R<x>:=PowerSeriesRing(Rationals(), 50);
%o A292915 T:= func< n,k | Coefficient(R!(Laplace( Exp(k*x)/(2-Exp(x)) )), n) >;
%o A292915 A292915:= func< n,k | T(k,n-k) >;
%o A292915 [A292915(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 12 2024
%o A292915 (SageMath)
%o A292915 def T(n,k): return factorial(n)*( exp(k*x)/(2-exp(x)) ).series(x, n+1).list()[n]
%o A292915 def A292915(n,k): return T(k,n-k)
%o A292915 flatten([[A292915(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jun 12 2024
%Y A292915 Columns k=0..4 give A000670, A000629, A007047, A259533, A368317.
%Y A292915 Rows n=0..2 give A000012, A000027, A102305.
%Y A292915 Main diagonal gives A292916.
%K A292915 nonn,tabl
%O A292915 0,5
%A A292915 _Ilya Gutkovskiy_, Sep 26 2017