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A060058 Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).

%I A060058 #25 May 04 2025 03:26:05
%S A060058 1,1,1,1,5,5,1,14,61,61,1,30,331,1385,1385,1,55,1211,12284,50521,
%T A060058 50521,1,91,3486,68060,663061,2702765,2702765,1,140,8526,281210,
%U A060058 5162421,49164554,199360981,199360981,1,204,18522,948002,28862471,510964090,4798037791,19391512145,19391512145
%N A060058 Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).
%H A060058 Wolfdieter Lang, <a href="/A060058/a060058.txt">First 9 rows</a>.
%F A060058 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 A060058 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 A060058 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 A060058 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 A060058 Triangle T(n, k) starts:
%e A060058   [0] 1;
%e A060058   [1] 1,   1;
%e A060058   [2] 1,   5,    5;
%e A060058   [3] 1,  14,   61,     61;
%e A060058   [4] 1,  30,  331,   1385,    1385;
%e A060058   [5] 1,  55, 1211,  12284,   50521,    50521;
%e A060058   [6] 1,  91, 3486,  68060,  663061,  2702765,   2702765;
%e A060058   [7] 1, 140, 8526, 281210, 5162421, 49164554, 199360981, 199360981;
%e A060058   ...
%p A060058 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 A060058 seq(print(seq(T(n, k), k=0..n)), n=0..7);  # _Peter Luschny_, Sep 30 2023
%t A060058 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 A060058 Cf. A060059 (row sums), A000364 (main diagonal Euler numbers).
%Y A060058 Columns: A000012 (powers of 1), A000330 (sum of squares), A060060-2 for m=0,...,4.
%Y A060058 See triangle A060074.
%K A060058 nonn,easy,tabl
%O A060058 0,5
%A A060058 _Wolfdieter Lang_, Mar 16 2001