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%I #55 Dec 19 2020 16:35:51
%S 1,1,1,1,4,1,1,13,10,1,1,40,73,20,1,1,121,478,273,35,1,1,364,2989,
%T 3208,798,56,1,1,1093,18298,35069,15178,1974,84,1,1,3280,110881,
%U 368988,262739,56632,4326,120,1,1,9841,668566,3800761,4310073,1452011,177760
%N Stirling-like number triangle defined by sequence A000217.
%C Columns include A003462, A016211, A021514. The defining sequence A000217(n) = C(n+1,2) is the sequence of partial sums of the sequence (0,1,2,3,4,...) which defines the Stirling numbers of the second kind A008277.
%C n-th row = M^n * [1,0,0,0,...], where M = an infinite lower triangular matrix with (1, 3, 6, ...) in the main diagonal and (1, 1, 1, ...) in the subdiagonal. - _Gary W. Adamson_, Apr 13 2009
%C Row sums = A124373 starting (1, 2, 6, 25, 135, ...). - _Gary W. Adamson_, Jul 11 2011
%H Vincenzo Librandi, <a href="/A080248/b080248.txt">Rows n = 0..50, flattened</a>
%F Columns are generated by 1/Product_{k=1..n+1} (1 - C(k + 1, 2)*x). [In other words:
%F T(n, k) = [x^(n-k)] 1/Product_{j=0..k+2}(1 - x*binomial(j, 2)).]
%F T(n, k) = (k*(k+1)/2) * T(n-1,k) + T(n-1,k-1), T(n,n)=1. - _Vladimir Kruchinin_, Aug 25 2020
%F T(n,k) = (Sum_{i=0..k} (-1)^(k-i) * (2*i + 3) * binomial(2*k + 3,k-i) * ((i+1) * (i+2) / 2)^(n+1)) * 2^(k+1) / (2*k + 3)! for 0 <= k <= n. - _Werner Schulte_, Oct 29 2020
%F The polynomials p(n,x) = Sum_{k=0..n} T(n,k) * (k!*(k+1)!/2^k) * x^(k+2) satisfy for n >= 0 the equations p(n+1,x) = p(1,x) * p''(n,x) / 2 and p(n,-1) = 0^n when p'' is the second derivative of p. - _Werner Schulte_, Dec 15 2020
%e Rows are
%e {1},
%e {1, 1},
%e {1, 4, 1},
%e {1, 13, 10, 1},
%e {1, 40, 73, 20, 1},
%e ...
%e For example, 73 = 13 + 6*10, 20 = 10 + 10*1.
%p gf := k -> 1/mul(1 - x*j*(j-1)/2, j=0..k+2):
%p ser := k -> series(gf(k), x, 16):
%p T := (n, k) -> coeff(ser(k), x, n-k):
%p seq(print(seq(T(n, k), k=0..n)), n=0..8); # _Peter Luschny_, Aug 29 2020
%t max = 10; t[n_, n_] = n*(n+1)/2; t[n_, k_] /; k == n-1 = 1; t[_, _] = 0; m = Table[t[n, k], {n, 1, max}, {k, 1, max}]; row[n_] := MatrixPower[m, n][[All, 1]]; Table[Take[row[n], n+1], {n, 0, max-1}] // Flatten (* _Jean-François Alcover_, Jun 25 2013, after _Gary W. Adamson_ *)
%o (PARI) {T(n, k) = local(s); if( k<0 || k>n, 0, forvec(v = vector(n-k, i, [0, k]), s += prod(i=1, n-k, v[i] * (v[i] + 1) / 2), 1)); s}; /* _Michael Somos_, Feb 06 2004 */
%Y Cf. A000217, A008277, A124373.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Feb 17 2003
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