A201595 E.g.f. satisfies A(x) = exp(x*A(x)) * cosh(x*A(x)).
1, 1, 4, 28, 288, 3936, 67328, 1385728, 33372160, 921118720, 28677169152, 994360565760, 38007586684928, 1587878686621696, 71990467473965056, 3520403893852831744, 184707311409882464256, 10350444842488122310656, 616975843658373414256640, 38981881007475178476666880
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! +... The coefficients of x^n/n! in initial powers of G(x) = (1 + exp(2*x))/2 begin: G^1: [(1), 1, 2, 4, 8, 16, 32, 64, 128, ...]; G^2: [1,(2), 6, 20, 72, 272, 1056, 4160, ...]; G^3: [1, 3,(12), 54, 264, 1368, 7392, 41184, ...]; G^4: [1, 4, 20,(112), 680, 4384, 29600, 207232, ...]; G^5: [1, 5, 30, 200,(1440), 11000, 88080, 732800, ...]; G^6: [1, 6, 42, 324, 2688,(23616), 217392, 2080224, ...]; G^7: [1, 7, 56, 490, 4592, 45472,(471296), 5076400, ...]; G^8: [1, 8, 72, 704, 7344, 80768, 928512,(11085824), ...]; ... where coefficients in parenthesis form initial terms of this sequence: [1/1, 2/2, 12/3, 112/4, 1440/5, 23616/6, 471296/7, 11085824/8, ...].
Links
- G. C. Greubel, Table of n, a(n) for n = 0..370
- Nantel Bergeron, Laura Colmenarejo, Shu Xiao Li, John Machacek, Robin Sulzgruber, Mike Zabrocki, Adriano Garsia, Marino Romero, Don Qui, and Nolan Wallach, Super Harmonics and a representation theoretic model for the Delta conjecture, A summary of the open problem sessions of Jan 24, 2019, Representation Theory Connections to (q,t)-Combinatorics (19w5131), Banff, BC, Canada.
- John Lentfer, A conjectural basis for the (1,2)-bosonic-fermionic coinvariant ring, arXiv:2406.19715 [math.CO], 2024. See p. 2.
Programs
-
Mathematica
Join[{1},Table[Sum[Binomial[n+1,k] k^n/(n+1),{k,0,n+1}]/2,{n,20}]] (* Harvey P. Dale, Feb 04 2012 *) CoefficientList[Series[(x-LambertW[-x*E^x])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 04 2012 *) -
PARI
a(n)=n!*polcoeff(1/x*serreverse(x*exp(-x+x^2*O(x^n))/cosh(x+x^2*O(x^n))),n)
-
PARI
a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp((n+1)*X)*cosh(X)^(n+1)/(n+1),n)
-
PARI
a(n)=sum(k=0,n+1,binomial(n+1,k)*k^n/(n+1)/2)
-
PARI
/* Formula for a(n,m) where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!: */ {a(n,m=1)=sum(k=0, n+m, binomial(n+m, k)*k^n*m/(n+m)/2^m)} -
PARI
/* Formula derived from a LambertW identity: */ {a(n)=local(A=sum(k=0,n,(k+1)^(k-1)*cosh((k+1)*x+x*O(x^n))*x^k/k!));n!*polcoeff(A,n)} for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Oct 24 2012 -
PARI
/* Formula derived from a LambertW identity: */ {a(n)=local(A=1+sum(k=1,n,k^k*sinh(k*x+x^2*O(x^n))/(k*x)*x^k/k!));n!*polcoeff(A,n)} for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012
Formula
a(n) = (1/2) * Sum_{k=0..n+1} C(n+1,k) * k^n / (n+1).
a(n) = [x^n/n!] exp((n+1)*x) * cosh(x)^(n+1) / (n+1).
E.g.f. A(x) satisfies:
(1) A( x*exp(-x)/cosh(x) ) = exp(x)*cosh(x).
(2) A(x) = (1/x)*Series_Reversion( x*exp(-x)/cosh(x) ).
(3) A(x) = (1 + exp(2*x*A(x)))/2.
(4) A(x) = exp(G(x)) where G(x) is the e.g.f. of A074932.
(5) A(x) = Sum_{n>=0} (n+1)^(n-1) * cosh((n+1)*x) * x^n/n!. - Paul D. Hanna, Oct 24 2012
(6) A(x) = 1 + Sum_{n>=1} n^n * sinh(n*x)/(n*x) * x^n/n!. - Paul D. Hanna, Nov 20 2012
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = Sum_{k=0..n+m} C(n+m, k) * k^n * m/(n+m) / 2^m.
a(n) = A214225(n+1)/(n+1).
E.g.f.: (x-LambertW(-x*exp(x)))/(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!*sqrt(LambertW(exp(-1))+1)/(2*sqrt(2*Pi)*n^(3/2)*LambertW(exp(-1))^(n+1)). - Vaclav Kotesovec, Dec 04 2012
G.f.: 1/2 + 1/2 * Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1). - Seiichi Manyama, Apr 23 2024
a(n) = n! * Sum_{k=0..n} 2^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024