cp's OEIS Frontend

A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)*binomial(n+m+1, n-k-i)*(n + m - k)!/(i + m)!, for m = 2.

%I A376634 #15 Oct 12 2024 15:37:52
%S A376634 1,9,1,71,12,1,580,119,15,1,5104,1175,179,18,1,48860,12154,2070,251,
%T A376634 21,1,509004,133938,24574,3325,335,24,1,5753736,1580508,305956,44524,
%U A376634 5000,431,27,1,70290936,19978308,4028156,617624,74524,7155,539,30,1,924118272,270074016,56231712,8969148,1139292,117454,9850,659,33,1,13020978816,3894932448,832391136,136954044,18083484,1961470,176554,13145,791,36,1
%N A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)*binomial(n+m+1, n-k-i)*(n + m - k)!/(i + m)!, for m = 2.
%C A376634 The columns of the triangle T(m,n,k) represent the coefficients of the asymptotic expansion of the higher order exponential integral E(x,m+1,k+2), for m=2, k>=0. For reference see. A163931.
%H A376634 Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-10-1.pdf#page=19">Relationships of P-generalized Stirling numbers of the first kind with other generalized Stirling numbers</a>, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.
%e A376634 Triangle starts:
%e A376634  [0]          1;
%e A376634  [1]          9,          1;
%e A376634  [2]         71,         12,          1;
%e A376634  [3]        580,        119,         15,        1;
%e A376634  [4]       5104,       1175,        179,       18,        1;
%e A376634  [5]      48860,      12154,       2070,      251,       21,       1;
%e A376634  [6]     509004,     133938,      24574,     3325,      335,      24,     1;
%e A376634  [7]    5753736,    1580508,     305956,    44524,     5000,     431,    27,     1;
%p A376634 T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k):m:=2:seq(seq(T(m,n,k), k=0..n),n=0..10);
%Y A376634 Column k: A001706 (k=0), A001712 (k=1), A001717 (k=2), A001722 (k=3), A051525 (k=4), A051546 (k=5), A051561 (k=6).
%Y A376634 Cf. A094587 and A173333 for m=0, A376582 for m=1.
%K A376634 nonn,tabl
%O A376634 0,2
%A A376634 _Igor Victorovich Statsenko_, Sep 30 2024