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Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).
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%I #38 Jun 04 2024 16:51:40

%S 1,2,5,16,66,348,2298,18504,176841,1958746,24661493,347548376,

%T 5415830272,92410046544,1712819553864,34258146124320,735267392077962,

%U 16852848083339700,410809882438699346,10611174406149372736,289493459925589039804,8317946739043065421640

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

%C 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 Vaclav Kotesovec, <a href="/A084785/b084785.txt">Table of n, a(n) for n = 0..350</a>

%H 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 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 G.f. A(x) satisfies (1+x)^2 = A(x/(1+x))^2/A(x). - _Michael Somos_, Feb 16 2006

%F G.f.: A(x) = Product_{n>=1} 1/(1 - n*x)^(1/2^n). - _Paul D. Hanna_, Jun 16 2010

%F a(n) ~ (n-1)! / (log(2))^(n+1). - _Vaclav Kotesovec_, Nov 19 2014

%F From _Peter Bala_, May 26 2001: (Start)

%F 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 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 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 nmax = 19; sol = {a[0] -> 1};

%t 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 sol /. Rule -> Set;

%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)

%t 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 (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 (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 (Magma)

%o m:=40;

%o 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 R<x>:=PowerSeriesRing(Rationals(), m+1); // A084785

%o Coefficients(R!( f(m,x) )); // _G. C. Greubel_, Jun 08 2023

%o (SageMath)

%o 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 m=50

%o def A084785_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P( f(m,x) ).list()

%o A084785_list(m-9) # _G. C. Greubel_, Jun 08 2023

%Y Cf. A000629, A000670, A019538, A084783, A084784, A084786, A090352.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Jun 13 2003

%E More terms from _David Wasserman_, Jan 06 2005