A156631 -id:A156631 - cp's OEIS frontend

A155200 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2) * x^n/n ), a power series in x with integer coefficients.

1, 2, 10, 188, 16774, 6745436, 11466849412, 80444398636280, 2306003967992402758, 268654794629082985019564, 126765597346260977505891041836, 241678070948246232010898235031930952, 1858395916567787793818891330877931472153500, 57560683587056536617649234722821582390470430186648

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, it appears that for m integer, exp( Sum_{n >= 1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.
This is correct: if b(n) = m^(n^2) then by the little Fermat theorem the Gauss congruences hold: b(n*p^k) == b(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k. Then apply Stanley, Ch. 5, Ex. 5.2(a). - Peter Bala, Dec 25 2019
Conjecture: highest exponent of 2 dividing a(n) = A000120(n) = number of 1's in binary expansion of n, so that a(n)/2^A000120(n) is odd for n >= 0. - Paul D. Hanna, Sep 01 2009

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 +...
log(A(x)) = 2*x + 2^4*x^2/2 + 2^9*x^3/3 + 2^16*x^4/4 + 2^25*x^5/5 +...

R. P. Stanley. Enumerative combinatorics, Vol. 2. Volume 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.

Muniru A Asiru, Table of n, a(n) for n = 0..56
Sawian Jaidee, Patrick Moss, and Tom Ward, Time-changes preserving zeta functions, arXiv:1809.09199 [math.DS], 2018.

Cf. A155201, A155202, A155810 (triangle), variants: A155203, A155207.

Cf. A000120, A156631, A159034.

seq(coeff(series(exp(add(2^(k^2)*x^k/k,k=1..n)),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Dec 19 2018

nmax = 14; Exp[Sum[2^(n^2) x^n/n, {n, 1, nmax}]] + O[x]^nmax // CoefficientList[#, x]& (* Jean-François Alcover, Feb 14 2019 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,2^(m^2)*x^m/m)+x*O(x^n)),n)}

{a(n)=if(n==0,1,(1/n)*(2*a(n-1) + sum(k=1,n-1,4^k*a(k)*(2*(k+1)*a(n-1-k) - (n-k)*a(n-k)))))} \\ Paul D. Hanna, Mar 11 2009

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,2^(k^2)*a(n-k)))} \\ Paul D. Hanna, Sep 01 2009

Equals column 0 of triangle A155810.
G.f. satisfies: 2*A(x)*A(4x) + 8*x*A(x)*A'(4x) - A'(x)*A(4x) = 0. - Paul D. Hanna, Feb 24 2009
From Paul D. Hanna, Mar 11 2009: (Start)
The differential equation implies recurrence:
n*a(n) = 2*a(n-1) + Sum_{k=1..n-1} 4^k*a(k)*(2*(k+1)*a(n-1-k) - (n-k)*a(n-k)) for n > 0, with a(0) = 1.
G.f. A(x) generates A156631:
A156631(n) = [x^n] A(x)^(2^n) for n >= 0, where the g.f. of A156631 = Sum_{n >= 0} [Sum_{k >= 1} (2^n*2^k*x)^k/k]^n/n!. (End)
a(n) = (1/n)*Sum_{k = 1..n} 2^(k^2)*a(n-k), a(0) = 1. - Vladeta Jovovic, Feb 04 2009
Euler transform of A159034. - Vladeta Jovovic, Apr 02 2009
a(n) = B_n( 0!*2^(1^2), 1!*2^(2^2), 2!*2^(3^2), ..., (n-1)!*2^(n^2) ) / n!, where B_n() is the complete Bell polynomial. - Max Alekseyev, Oct 10 2014
a(n) ~ 2^(n^2) / n. - Vaclav Kotesovec, Oct 09 2019

A156630 G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} (2^n + 2^k)^k*x^k/k ]^n / n!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 36, 692, 38186, 10012732, 14013453284, 89892733239928, 2455110210935634790, 278266942487534934333100, 129264916198375365693754194988, 244287539590735476133066282560012360

Offset: 0

Paul D. Hanna, Feb 12 2009

nonn

Compare to these dual g.f.s:
Sum_{n>=0} [ Sum_{k>=1} (2^n+1)^k*x^k/k ]^n/n! (A133991);
Sum_{n>=0} [ Sum_{k>=1} (2^k+1)^k*x^k/k ]^n/n! (A155201);
which, when expanded as power series in x, have only integer coefficients.

			G.f.: A(x) = 1 + 4*x + 36*x^2 + 692*x^3 + 38186*x^4 + 10012732*x^5 +...

Cf. A156631, A133991, A155201.

{a(n)=polcoeff(sum(j=0,n,sum(k=1, n, ((2^j+2^k)*x)^k/k+x*O(x^n))^j/j!),n)}

A164764 G.f. satisfies: the coefficient of x^n in A(x)^n = 2^(n^2) for n>=1 with A(0)=1.

Original entry on oeis.org

1, 2, 6, 144, 15390, 6580224, 11386265292, 80284132772352, 2304717583810291830, 268613293782939614576640, 126760224526971269877523841364, 241675282146473482949215936098066432

Offset: 0

Paul D. Hanna, Aug 25 2009

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 144*x^3 + 15390*x^4 + 6580224*x^5 +...
The coefficients in the successive powers of A(x) begin:
[1,(2), 6, 144, 15390, 6580224, 11386265292, 80284132772352,...];
[1, 4,(16), 312, 31392, 13223736, 22799056896, 160613894000880,...];
[1, 6, 30,(512), 48078, 19932480, 34238570076, 240989363896320,...];
[1, 8, 48, 752,(65536), 26708592, 45705005568, 321410623205088,...];
[1, 10, 70, 1040, 83870,(33554432), 57198570060, 401877753223680,...];
[1, 12, 96, 1384, 103200, 40472616,(68719476736), 482390835814224,...];
[1, 14, 126, 1792, 123662, 47466048, 80267945884,(562949953421312),...];
[1, 16, 160, 2272, 145408, 54537952, 91844205568, 643555189090240,...];
...
The above terms in parenthesis = [x^n] A(x)^n = 2^(n^2) for n=1,2,3,...
The main diagonal = [x^n] A(x)^(n+1) = (n+1)*A155200(n):
[1, 2*2, 3*10, 4*188, 5*16774, 6*6745436, 7*11466849412, ...].

Cf. A155200, A156214, A156631.

{a(n)=local(G=exp(sum(m=1,n,2^(m^2)*x^m/m)+x*O(x^n)));polcoeff(x/serreverse(x*G),n)}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,2^(m^2)*(x/A)^m/m)));polcoeff(A,n)}

G.f. satisfies: A(x) = exp( Sum_{n>=1} 2^(n^2)*[x/A(x)]^n/n ).
Let G(x) = exp(Sum_{n>=1} 2^(n^2)*x^n/n) = g.f. of A155200, then:
(1) A(x) = G(x/A(x)) and A(x*G(x)) = G(x) ;
(2) A(x) = x/Series_Reversion[x*G(x)] ;
(3) [x^n] A(x)^(n+1)/(n+1) = [x^n] G(x) = A155200(n) ;
(4) [x^n] A(x)^(n+m)*m/(n+m) = [x^n] G(x)^m for all m.

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A155200 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2) * x^n/n ), a power series in x with integer coefficients.

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A156630 G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} (2^n + 2^k)^k*x^k/k ]^n / n!, a power series in x with integer coefficients.

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A164764 G.f. satisfies: the coefficient of x^n in A(x)^n = 2^(n^2) for n>=1 with A(0)=1.

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