cp's OEIS Frontend

A084785 Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).

%I A084785 #38 Jun 04 2024 16:51:40
%S A084785 1,2,5,16,66,348,2298,18504,176841,1958746,24661493,347548376,
%T A084785 5415830272,92410046544,1712819553864,34258146124320,735267392077962,
%U A084785 16852848083339700,410809882438699346,10611174406149372736,289493459925589039804,8317946739043065421640
%N A084785 Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).
%C A084785 In the triangle (A084783), the diagonal (this sequence) is the self-convolution of the first column (A084784) and the row sums (A084786) gives the differences of the diagonal and the first column.
%H A084785 Vaclav Kotesovec, <a href="/A084785/b084785.txt">Table of n, a(n) for n = 0..350</a>
%H A084785 Chao-Ping Chen, <a href="https://doi.org/10.1016/j.jnt.2016.08.010">Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant</a>, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159.
%H A084785 Olivier Golinelli, <a href="https://arxiv.org/abs/2405.16968">Remote control system of a binary tree of switches - II. balancing for a perfect binary tree</a>, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
%F A084785 G.f. A(x) satisfies (1+x)^2 = A(x/(1+x))^2/A(x). - _Michael Somos_, Feb 16 2006
%F A084785 G.f.: A(x) = Product_{n>=1} 1/(1 - n*x)^(1/2^n). - _Paul D. Hanna_, Jun 16 2010
%F A084785 a(n) ~ (n-1)! / (log(2))^(n+1). - _Vaclav Kotesovec_, Nov 19 2014
%F A084785 From _Peter Bala_, May 26 2001: (Start)
%F A084785 O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-2)^k = A000629(n) = 2*A000670(n) for n >= 1. Cf. A090352.
%F A084785 sqrt(A(x)) = 1/(1 + x)*A(x/(1 + x)) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 137*x^5 + ... is the o.g.f. for A084784. See also A019538. (End)
%e A084785 G.f.: A(x) = (1-x)^(-1/2)*(1-2*x)^(-1/4)*(1-3*x)^(-1/8)*(1-4*x)^(-1/16)*... - _Paul D. Hanna_, Jun 16 2010
%t A084785 nmax = 19; sol = {a[0] -> 1};
%t A084785 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[(1+x)^2 * A[x] - A[x/(1+x)]^2 + O[x]^(n+1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A084785 sol /. Rule -> Set;
%t A084785 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)
%t A084785 With[{m=40}, CoefficientList[Series[Exp[Sum[Sum[(-2)^j*j!*StirlingS2[k, j], {j,k}]*(-x)^k /k, {k,m+1}]], {x,0,m}], x]] (* _G. C. Greubel_, Jun 08 2023 *)
%o A084785 (PARI) A = matrix(25, 25); A[1, 1] = 1; rs = 1; print(1); for (n=2, 25, sc = sum(i=2, n-1, A[i, 1]*A[n+1-i, 1]); A[n, 1] = rs - sc; rs = A[n, 1]; for (k=2, n, A[n, k] = A[n, k-1] + A[n-1, k-1]; rs += A[n, k]); print(A[n, n])); \\ _David Wasserman_, Jan 06 2005
%o A084785 (PARI) {a(n)=local(A); if(n<0, 0, A=1; for(k=1,n, A=truncate(A+O(x^k))+x*O(x^k); A+=A-(subst(1/A,x,x/(1+x))*(1+x))^-2;); polcoeff(A,n))} /* _Michael Somos_, Feb 18 2006 */
%o A084785 (Magma)
%o A084785 m:=40;
%o A084785 f:= func< n,x | Exp((&+[(&+[(-2)^j*Factorial(j)*StirlingSecond(k,j)*(-x)^k/k: j in [1..k]]): k in [1..n+2]])) >;
%o A084785 R<x>:=PowerSeriesRing(Rationals(), m+1);  // A084785
%o A084785 Coefficients(R!( f(m,x) )); // _G. C. Greubel_, Jun 08 2023
%o A084785 (SageMath)
%o A084785 def f(n, x): return exp(sum(sum( (-2)^j*factorial(j)* stirling_number2(k,j)*(-x)^k/k for j in range(1,k+1)) for k in range(1,n+2)))
%o A084785 m=50
%o A084785 def A084785_list(prec):
%o A084785     P.<x> = PowerSeriesRing(QQ, prec)
%o A084785     return P( f(m,x) ).list()
%o A084785 A084785_list(m-9) # _G. C. Greubel_, Jun 08 2023
%Y A084785 Cf. A000629, A000670, A019538, A084783, A084784, A084786, A090352.
%K A084785 nonn,easy
%O A084785 0,2
%A A084785 _Paul D. Hanna_, Jun 13 2003
%E A084785 More terms from _David Wasserman_, Jan 06 2005