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

0,2

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.

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.

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)

Example

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

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 *)

Prog

(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 */
(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<x>:=PowerSeriesRing(Rationals(), m+1);  // A084785
Coefficients(R!( f(m, x) )); // G. C. Greubel, Jun 08 2023
(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.<x> = PowerSeriesRing(QQ, prec)
    return P( f(m, x) ).list()
A084785_list(m-9) # G. C. Greubel, Jun 08 2023

Crossrefs

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

Sequence in context: A007469 A306026 A091139 * A124551 A331157 A005157

Adjacent sequences: A084782 A084783 A084784 * A084786 A084787 A084788

Keyword

nonn,easy

Author

Paul D. Hanna, Jun 13 2003

Extensions

More terms from David Wasserman, Jan 06 2005

Status

approved