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%I #11 Oct 30 2021 07:17:03
%S 1,1,1,1,2,2,1,4,6,5,1,7,18,23,16,1,11,46,95,108,65,1,16,101,325,583,
%T 605,326,1,22,197,931,2533,4103,3956,1957,1,29,351,2310,9050,21834,
%U 32677,29649,13700,1,37,583,5118,27530,94234,207349,291065,250892,109601,1,46,916,10365,73592,342004,1055455,2157206,2870477,2367629,986410
%N Triangle generated by the recurrence T(n+1,k+1) = T(n,k+1) + n * T(n,k) + delta(n,k) with the initial values T(n,0) = 1 and T(0,k) = delta(k,0), where delta(n,k) is the Kronecker delta.
%C Row sums = A000522.
%C Diagonal sums = A191491.
%C Central coefficients = A191492.
%C Binomial row sums = A191493.
%C Let r(n) = sum(T(n,k),k=0..n) be the row sums.
%C Let s(n) = sum(T(n,k)*(-1)^(n-k),k=0..n) be the alternated row sums.
%C Let d(n) = T(n,n) be the diagonal elements.
%C Then s(n+2) = r(n) and r(n) = d(n+1).
%H Vincenzo Librandi, <a href="/A191490/b191490.txt">Table of n, a(n) for n = 0..560</a>
%F Recurrence: T(n+1,k+1) = sum(i*T(i,k),i=0..n)+[k<=n],
%F where [k<=n]=1 if k<=n and [k<=n]=0 if k>n.
%F Mixed generating series:
%F sum(T(n,k)*q^k*x^n/n!,n=0..inf) = (1-q*x)^(-1/q)*(1+q*int(exp(q*t)/(1-q*t)^((q-1)/q),t=0..x)).
%F Let f(n,q)= sum(T(n,k)*q^k,k=0..n) the generating polynomials of the rows. Then f(n+1,q)=(1+n*q)*f(n,q)+q^(n+1).
%F Let A(n,q)=sum(s(n,n-k)*q^k,k=0..n), where the coefficients s(n,k) are the (signless) Stirling numbers of the first kind.
%F Let B(n,q)=sum(sum(binomial(n-1,i)*s(n-i-1,k),i=0..n-1)*(q-1)^k*q^(n-k),k=0..n-1). Finally, let P(n,q)=A(n,q)+sum(binomial(n,k)*A(k,q)*B(n-k,q),k=0..n). Then T(n,k)=[q^k]P(n,q).
%e Triangle begins:
%e 1
%e 1, 1
%e 1, 2, 2
%e 1, 4, 6, 5
%e 1, 7, 18, 23, 16
%e 1, 11, 46, 95, 108, 65
%e 1, 16, 101, 325, 583, 605, 326
%e 1, 22, 197, 931, 2533, 4103, 3956, 1957
%e 1, 29, 351, 2310, 9050, 21834, 32677, 29649, 13700
%t f[n_, k_] := f[n, k] = f[n - 1, k] + (n - 1)f[n - 1, k - 1] + If[n == k, 1, 0]
%t f[_, 0] = 1;
%t f[0, _] = 0;
%t Flatten[Table[f[n, k], {n, 0, 100}, {k, 0, n}]]
%o (Maxima) P[0]:1$
%o P[n]:=(1+(n-1)*q)*P[n-1]+q^n$
%o create_list(coeff(expand(P[n]),q^k),n,0,12,k,0,n);
%Y Cf. A000522, A191491, A191492, A191493.
%K nonn,tabl
%O 0,5
%A _Emanuele Munarini_, Jun 03 2011
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