cp's OEIS Frontend

A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

%I A325872 #30 Apr 18 2025 09:52:49
%S A325872 1,0,1,0,-2,1,0,7,-6,1,0,-35,40,-12,1,0,228,-315,130,-20,1,0,-1834,
%T A325872 2908,-1485,320,-30,1,0,17582,-30989,18508,-5005,665,-42,1,0,-195866,
%U A325872 375611,-253400,81088,-13650,1232,-56,1,0,2487832,-5112570,3805723,-1389612,279048,-32130,2100,-72,1
%N A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.
%H A325872 Michael De Vlieger, <a href="/A325872/b325872.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H A325872 Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, <a href="https://doi.org/10.1515/gmj-2019-2007">A new set of Sheffer-Bell polynomials and logarithmic numbers</a>, Georgian Mathematical Journal, Feb. 2019, page 8.
%H A325872 Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%F A325872 From _Seiichi Manyama_, Apr 18 2025: (Start)
%F A325872 T(n,k) = Sum_{j=k..n} Stirling1(n,j) * Stirling1(j,k).
%F A325872 E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + log(1 + x)). (End)
%e A325872 Triangle starts:
%e A325872 [0] [1]
%e A325872 [1] [0,       1]
%e A325872 [2] [0,      -2,        1]
%e A325872 [3] [0,       7,       -6,       1]
%e A325872 [4] [0,     -35,       40,     -12,        1]
%e A325872 [5] [0,     228,     -315,     130,      -20,      1]
%e A325872 [6] [0,   -1834,     2908,   -1485,      320,    -30,      1]
%e A325872 [7] [0,   17582,   -30989,   18508,    -5005,    665,    -42,    1]
%e A325872 [8] [0, -195866,   375611, -253400,    81088, -13650,   1232,  -56,   1]
%e A325872 [9] [0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1]
%t A325872 p[n_] := Sum[StirlingS1[n, k] FactorialPower[x, k] , {k, 0, n}];
%t A325872 Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten
%o A325872 (Sage)
%o A325872 def a_row(n):
%o A325872     s = sum((-1)^(n-k)*stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
%o A325872     return expand(s).list()
%o A325872 [a_row(n) for n in (0..9)]
%o A325872 (PARI) T(n, k) = sum(j=k, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ _Seiichi Manyama_, Apr 18 2025
%Y A325872 Columns k=0..3 give A000007, (-1)^(n+1) * A003713(n), (-1)^n * A341587(n), (-1)^(n+1) * A341588(n).
%Y A325872 Cf. A039814 (variant), A129062, A325873.
%K A325872 sign,tabl
%O A325872 0,5
%A A325872 _Peter Luschny_, Jun 27 2019