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Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).
16

%I #20 Sep 30 2023 09:19:54

%S 1,1,1,1,5,5,1,14,61,61,1,30,331,1385,1385,1,55,1211,12284,50521,

%T 50521,1,91,3486,68060,663061,2702765,2702765,1,140,8526,281210,

%U 5162421,49164554,199360981,199360981,1,204

%N Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).

%H W. Lang, <a href="/A060058/a060058.txt">First 9 rows</a>.

%F a(n, m) = a(n-1, m) + ((n+1-m)^2)*a(n, m-1), a(n, -1) := 0, a(0, 0) = 1, a(n, m) = 0 if n < m.

%F a(n, m) = ay(n-m+1, m) if n >= m >= 0, with the rectangular array ay(n, m) := Sum_{j=1..n} (j^2)*ay(j+1, m-1), n >= 0, m >= 1; input: ay(n, 0)=1 (iterated sums of squares).

%F G.f. for m-th column: 1/(1-x) for m=0, (x^m)*(Sum_{k=0..m} A060063(m, k)*x^k)/(1-x)^(3*m+1), m >= 1.

%F Recursion for g.f.s for m-th column: (1-x)*G(m, x) = x*G''(m-1, x) - G'(m-1, x) + G(m-1, x)/x, m >= 2; G(1, x) = x*(1+x)/(1-x)^4; the apostrophe denotes differentiation w.r.t. x. G(0, x) = 1/(1-x). - _Wolfdieter Lang_, Feb 13 2004.

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 5, 5;

%e [3] 1, 14, 61, 61;

%e [4] 1, 30, 331, 1385, 1385;

%e [5] 1, 55, 1211, 12284, 50521, 50521;

%e [6] 1, 91, 3486, 68060, 663061, 2702765, 2702765;

%e [7] 1, 140, 8526, 281210, 5162421, 49164554, 199360981, 199360981;

%p T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1) else (n - k + 1)^2 * T(n, k - 1) + T(n - 1, k) fi fi end:

%p seq(print(seq(T(n, k), k=0..n)), n=0..7); # _Peter Luschny_, Sep 30 2023

%t a[_, -1] = 0; a[0, 0] = 1; a[n_, m_] /; n < m = 0; a[n_, m_] := a[n, m] = a[n-1, m] + (n+1-m)^2*a[n, m-1]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 09 2013 *)

%Y Cf. A060059 (row sums), A000364 (main diagonal Euler numbers).

%Y Columns: A000012 (powers of 1), A000330 (sum of squares), A060060-2 for m=0,...,4.

%Y See triangle A060074.

%K nonn,easy,tabl

%O 0,5

%A _Wolfdieter Lang_, Mar 16 2001