A155207 -id:A155207 - 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

A155203 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 45, 6687, 10782369, 169490304819, 25016281429306077, 34185693516532070487615, 429210580094546346191627404353, 49269611092414945570325157106493868771

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, for m integer, exp( Sum_{n>=1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.

			G.f.: A(x) = 1 + 3*x + 45*x^2 + 6687*x^3 + 10782369*x^4 + 169490304819*x^5 +...
log(A(x)) = 3*x + 3^4*x^2/2 + 3^9*x^3/3 + 3^16*x^4/4 + 3^25*x^5/5 +...

Cf. A060722, A155204, A155205, A155206, A155812 (triangle), variants: A155200, A155207.

{a(n)=polcoeff(exp(sum(m=1,n+1,3^(m^2)*x^m/m)+x*O(x^n)),n)}

Equals column 0 of triangle A155812.
G.f. satisfies: A'(x)/A(x) = 3 + 27*x*A'(9*x)/A(9*x). - Paul D. Hanna, Nov 15 2022
a(n) ~ 3^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A155208 G.f.: A(x) = exp( Sum_{n>=1} (4^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 5, 157, 92285, 1091087581, 226287110093405, 788215837483128170845, 45292586018794926904179045725, 42540488665745908362239138191829777245, 649578584556365450465861374646071307864262693725

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

			G.f.: A(x) = 1 + 5*x + 157*x^2 + 92285*x^3 + 1091087581*x^4 +...
log(A(x)) = 5*x + 17^2*x^2/2 + 65^3*x^3/3 + 257^4*x^4/4 + 1025^5*x^5/5 +...

Cf. A155207, A155209, A155210, variants: A155201, A155204.

{a(n)=polcoeff(exp(sum(m=1,n+1,(4^m+1)^m*x^m/m)+x*O(x^n)),n)}

A155209 G.f.: A(x) = exp( Sum_{n>=1} (4^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 117, 83691, 1057319541, 224085796087563, 785909534807110163445, 45253898808490419883694669835, 42530103981310660908750359650219091445, 649533982980850199063905669772208004250784346635

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

			G.f.: A(x) = 1 + 3*x + 117*x^2 + 83691*x^3 + 1057319541*x^4 +...
log(A(x)) = 3*x + 15^2*x^2/2 + 63^3*x^3/3 + 255^4*x^4/4 + 1023^5*x^5/5 +...

Cf. A155207, A155208, A155210, A241098; variants: A155202, A155205.

{a(n)=polcoeff(exp(sum(m=1,n+1,(4^m-1)^m*x^m/m)+x*O(x^n)),n)}

A155210 G.f.: A(x) = exp( Sum_{n>=1} (4^n - 1)^n/3^(n-1) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 42, 9378, 39179127, 2766569881269, 3234201150559172040, 62076685218110095082936700, 19446778350632942283719042004313725, 98999235365955012033013202024947235500115415

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, for m integer, exp( Sum_{n>=1} (m^n - 1)^n/(m-1)^(n-1) * x^n/n ) is a power series in x with integer coefficients.

Note that g.f. exp( Sum_{n>=1} (4^n - 1)^n/3^n * x^n/n ) has fractional coefficients as a power series in x.

			G.f.: A(x) = 1 + 3*x + 42*x^2 + 9378*x^3 + 39179127*x^4 +...
log(A(x)) = 3*x + 15^2/3*x^2/2 + 63^3/3^2*x^3/3 + 255^4/3^3*x^4/4 +...

Cf. A155207, A155208, A155209, variant: A155206.

{a(n)=polcoeff(exp(sum(m=1,n+1,(4^m-1)^m/3^(m-1)*x^m/m)+x*O(x^n)),n)}

A156337 G.f.: A(x) = exp( Sum_{n>=1} 4^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 16, 384, 17856, 13492992, 11507268608, 160888878129152, 2306486569154275328, 537309590223329223155712, 126767209261235580163634135040, 483356141899716284828508078471905280

Offset: 0

Paul D. Hanna, Feb 10 2009

nonn

It appears that g.f. exp( Sum_{n>=1} m^[(n^2+1)/2]*x^n/n ) forms a power series in x with integer coefficients for any positive integer m.

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 384*x^3 + 17856*x^4 + 13492992*x^5 +...
log(A(x)) = 4*x + 4^2*x^2/2 + 4^5*x^3/3 + 4^8*x^4/4 + 4^13*x^5/5 + 4^18*x^6/6 +...

Cf. A156335, A156336, A155207, A155200.

{a(n)=polcoeff(exp(sum(k=1, n, 4^floor((k^2+1)/2)*x^k/k)+x*O(x^n)), n)}

a(n) = (1/n)*Sum_{k=1..n} 4^floor((k^2+1)/2) * a(n-k) for n>0, with a(0)=1.

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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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A155203 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2) * x^n/n ), a power series in x with integer coefficients.

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A155208 G.f.: A(x) = exp( Sum_{n>=1} (4^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

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A155209 G.f.: A(x) = exp( Sum_{n>=1} (4^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

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A155210 G.f.: A(x) = exp( Sum_{n>=1} (4^n - 1)^n/3^(n-1) * x^n/n ), a power series in x with integer coefficients.

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A156337 G.f.: A(x) = exp( Sum_{n>=1} 4^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

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