A158884 -id:A158884 - cp's OEIS frontend

A238223 Decimal expansion of a constant related to A088716.

2, 1, 7, 9, 5, 0, 7, 8, 9, 4, 4, 7, 1, 5, 1, 0, 6, 5, 4, 9, 2, 8, 2, 2, 8, 2, 2, 4, 4, 2, 3, 1, 9, 8, 2, 0, 8, 8, 6, 6, 0, 4, 5, 3, 9, 5, 6, 2, 9, 3, 9, 9, 6, 3, 4, 8, 1, 2, 3, 4, 0, 1, 7, 6, 2, 6, 5, 8, 7, 3, 3, 6, 2, 9, 2, 5, 3, 7, 0, 9, 4, 4, 9, 1, 2, 5, 9, 6, 3, 2, 2, 9, 8, 6, 2, 2, 9, 4, 5, 1, 4, 4, 8, 8, 9, 0

Offset: 0

Vaclav Kotesovec, Feb 21 2014

			0.21795078944715106549282282244231982088...

Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 13.

Cf. A088716, A088715, A158884, A159606, A182962, A193332, A156326, A338377.

Equals lim n->infinity A088716(n)/(n!*n^2).

A158883 G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^n for n>1 with A(0)=A'(0)=1.

Original entry on oeis.org

1, 1, -2, 9, -56, 425, -3726, 36652, -397440, 4695489, -59941550, 821711605, -12037503384, 187689245588, -3104186515976, 54295661153700, -1001685184237056, 19444296845046033, -396260414466644574, 8460628832978195683, -188898511962856879400

Offset: 0

Paul D. Hanna, Apr 30 2009

sign

			G.f.: A(x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 + ...
(d/dx) (x/A(x)) = 1 - 2*x + 9*x^2 - 56*x^3 + 425*x^4 - 3726*x^5 + ...
1/A(x) = 1 - x + 3*x^2 - 14*x^3 + 85*x^4 + ... + (-1)^n*A088716(n)*x^n + ...
where a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
...
Coefficients of powers of g.f. A(x) begin:
A^1: 1,1,-2,9,-56,425,-3726,36652,-397440,4695489,...;
A^2: 1,2,(-3),14,-90,702,-6297,63144,-695886,8334822,...;
A^3: 1,3,(-3),(16),-108,870,-7997,81774,-915798,11116902,...;
A^4: 1,4,-2,(16),(-115),960,-9050,94368,-1073658,13204560,...;
A^5: 1,5,0,15,(-115),(996),-9630,102365,-1182690,14730890,...;
A^6: 1,6,3,14,-111,(996),(-9870),106890,-1253466,15804548,...;
A^7: 1,7,7,14,-105,973,(-9870),(108816),-1294412,16514162,...;
A^8: 1,8,12,16,-98,936,-9704,(108816),(-1312227),16931984,...;
A^9: 1,9,18,21,-90,891,-9426,107406,(-1312227),(17116900),...;
A^10:1,10,25,30,-80,842,-9075,104980,-1298625,(17116900),...; ...
where coefficients [x^n] A(x)^(n+1) and [x^n] A(x)^n are
enclosed in parenthesis and equal (n+1)*A158884(n) for n > 1:
[ -3,16,-115,996,-9870,108816,-1312227,17116900,...];
compare to A158884:
[1,1,-1,4,-23,166,-1410,13602,-145803,1711690,-21785618,...]
and also to the logarithmic derivative of A158884:
[1,-3,16,-115,996,-9870,108816,-1312227,17116900,...].

Alois P. Heinz, Table of n, a(n) for n = 0..448

Cf. A088716, A158884, variant: A158882.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(j)*b(n-j-1)*(j+1), j=0..n-1))
    end:
a:= n-> `if`(n=0, 1, -(-1)^n*n*b(n-1)):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 18 2020

m = 19; A[_] = 1;
Do[A[x_] = 1 + x*D[x/A[x], x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *)

Composita(n,k,F):=if k=1 then F(n) else sum(F(i+1)*Composita(n-i-1,k-1,F),i,0,n-k);
array(a, 10);
a[1]:1;
af(n):=a[n];
for n:2 thru 10 do a[n]:n*sum(Composita(n-1, k, af)*(-1)^k , k, 1, n-1);
makelist(af(n),n,1,10); /* Vladimir Kruchinin, Dec 01 2011 */

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);A[n+1]}

G.f. satisfies: A(x) = 1 + x*(d/dx)(x/A(x)) so that x^2*A'(x) = x*A(x) + A(x)^2 - A(x)^3.
a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
G.f.: A(x) = 1/(Sum_{n>=0} (-1)^n*A088716(n)*x^n), where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*(d/dx)(x*F(x)).
G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)*A158884(n) for n > 1.

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A238223 Decimal expansion of a constant related to A088716.

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A158883 G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^n for n>1 with A(0)=A'(0)=1.

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