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A387152 Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..n} binomial(k, j)*|Stirling1(n, j)|.

%I A387152 #11 Aug 28 2025 17:26:34
%S A387152 1,0,1,0,1,1,0,1,2,1,0,2,3,3,1,0,6,7,6,4,1,0,24,23,16,10,5,1,0,120,98,
%T A387152 57,30,15,6,1,0,720,514,257,115,50,21,7,1,0,5040,3204,1407,546,205,77,
%U A387152 28,8,1,0,40320,23148,9076,3109,1021,336,112,36,9,1
%N A387152 Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..n} binomial(k, j)*|Stirling1(n, j)|.
%F A387152 T(n, k) = n! * [x^n] Laguerre(k, log(1 - x)).
%F A387152 From _Natalia L. Skirrow_, Aug 27 2025: (Start)
%F A387152 D-finite with T(n,k) = (n-1)*T(n-1,k)+T(n,k-1)-(n-2)*T(n-1,k-1).
%F A387152 O.g.f.: hypergeom([1,y/(1-y)],[],x)/(1-y).
%F A387152 Row o.g.f.: (y/(1-y))_n/(1-y), where (x)_n is the Pochhammer symbol/rising factorial.
%F A387152 Row o.g.f. is also 0^n + y/(1-y)^(n+1)*Prod_{j=1..n-2}(j+1-j*y).
%F A387152 E.g.f.: 1/((1-y)*(1-x)^(y/(1-y))).
%F A387152 Column e.g.f.: hypergeom([-k],[1],log(1-y)).
%F A387152 T(n,k) = [x^k] (1+x)^k*(x)_n.
%F A387152 (End)
%e A387152 Array begins:
%e A387152   [0]  1,     1,      1,      1,       1,       1,       1, ...
%e A387152   [1]  0,     1,      2,      3,       4,       5,       6, ...
%e A387152   [2]  0,     1,      3,      6,      10,      15,      21, ...
%e A387152   [3]  0,     2,      7,     16,      30,      50,      77, ...
%e A387152   [4]  0,     6,     23,     57,     115,     205,     336, ...
%e A387152   [5]  0,    24,     98,    257,     546,    1021,    1750, ...
%e A387152   [6]  0,   120,    514,   1407,    3109,    6030,   10696, ...
%e A387152   [7]  0,   720,   3204,   9076,   20695,   41330,   75356, ...
%e A387152   [8]  0,  5040,  23148,  67456,  157865,  323005,  602517, ...
%e A387152   [9]  0, 40320, 190224, 567836, 1358564, 2837549, 5396650, ...
%p A387152 A := (n, k) -> add(binomial(k, j)*abs(Stirling1(n, j)), j = 0..n):
%p A387152 seq(seq(A(n-k, k), k = 0..n), n = 0..10);
%p A387152 # Expanding rows or columns:
%p A387152 RowSer := n -> series((1+x)^k*GAMMA(x + n)/GAMMA(x), x, 12):
%p A387152 Trow := n -> k -> coeff(RowSer(n), x, k):
%p A387152 ColSer := n -> series(orthopoly:-L(n, log(1 - x)), x, 12):
%p A387152 Tcol := k -> n -> n! * coeff(ColSer(k), x, n):
%p A387152 seq(lprint(seq(Trow(n)(k), k = 0..7)), n = 0..9);
%p A387152 seq(lprint(seq(Tcol(k)(n), n = 0..7)), k = 0..9);
%o A387152 (Python)
%o A387152 from functools import cache
%o A387152 @cache
%o A387152 def T(n: int, k: int) -> int:
%o A387152     if n == 0: return 1
%o A387152     if k == 0: return 0
%o A387152     return (n - 1) * T(n - 1, k) + T(n, k - 1) - (n - 2) * T(n - 1, k - 1)
%o A387152 for n in range(7): print([T(n, k) for k in range(7)])
%Y A387152 Rows: A000012 [0], A001477 [1], A000217 [2], A005581 [3], A387204 [4].
%Y A387152 Columns: A000007 [0], A000142 [shifted, 1], A387205 [2].
%Y A387152 Contains A271700 in transpose.
%Y A387152 Cf. A211210 (main diagonal), A130534.
%K A387152 nonn,tabl,new
%O A387152 0,9
%A A387152 _Peter Luschny_, Aug 27 2025