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

1, 2, 5, 16, 66, 348, 2298, 18504, 176841, 1958746, 24661493, 347548376, 5415830272, 92410046544, 1712819553864, 34258146124320, 735267392077962, 16852848083339700, 410809882438699346, 10611174406149372736, 289493459925589039804, 8317946739043065421640

Offset: 0

Author

Paul D. Hanna, Jun 13 2003

Keywords

• nonn
• easy

Comments

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.

Examples

			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
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..350
• Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159.
• Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.

Crossrefs

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

Programs

Magma

m:=40;
f:= func< n,x | Exp((&+[(&+[(-2)^j*Factorial(j)*StirlingSecond(k,j)*(-x)^k/k: j in [1..k]]): k in [1..n+2]])) >;
R:=PowerSeriesRing(Rationals(), m+1);  // A084785
Coefficients(R!( f(m,x) )); // G. C. Greubel, Jun 08 2023

Mathematica

nmax = 19; sol = {a[0] -> 1};
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}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)
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 *)

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

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 */

SageMath

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)))
m=50
def A084785_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(m,x) ).list()
A084785_list(m-9) # G. C. Greubel, Jun 08 2023

Formula

G.f. A(x) satisfies (1+x)^2 = A(x/(1+x))^2/A(x). - Michael Somos, Feb 16 2006

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

a(n) ~ (n-1)! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 19 2014

From Peter Bala, May 26 2001: (Start)

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.

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)

Extensions

More terms from David Wasserman, Jan 06 2005