A155203 -id:A155203 - 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

A155204 G.f.: A(x) = exp( Sum_{n>=1} (3^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 58, 7528, 11333974, 173018964568, 25223063625377572, 34295288559321731710864, 429734241619476967064512081894, 49292144502053186639397817183561560472

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 + 4*x + 58*x^2 + 7528*x^3 + 11333974*x^4 + 173018964568*x^5 +...
log(A(x)) = 4*x + 10^2*x^2/2 + 28*x^3/3 + 82^4*x^4/4 + 244^5*x^5/5 +...

Cf. A155203, A155205, A155206, A155812 (triangle), A202989; variants: A155201, A155208.

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

Equals row sums of triangle A155812.

A155205 G.f.: A(x) = exp( Sum_{n>=1} (3^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 34, 5924, 10252294, 166020197708, 24810918565918804, 34076399079565985138408, 428687477154543524080261047622, 49247086840315416213775472777558582540

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 + 2*x + 34*x^2 + 5924*x^3 + 10252294*x^4 +...
log(A(x)) = 2*x + 8^2*x^2/2 + 26^3*x^3/3 + 80^4*x^4/4 + 242^5*x^5/5 +...

Cf. A060613, A155203, A155204, A155206, A155812 (triangle), variants: A155202, A155209.

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

A155207 G.f.: A(x) = exp( Sum_{n>=1} 4^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 136, 87904, 1074100576, 225184288253824, 787061981348092400896, 45273238870711805132010916864, 42535296046210357883346895894694749696, 649556283428320264374891976653586736162144180224

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 + 4*x + 136*x^2 + 87904*x^3 + 1074100576*x^4 +...
log(A(x)) = 4*x + 4^4*x^2/2 + 4^9*x^3/3 + 4^16*x^4/4 + 4^25*x^5/5 +...

Cf. A060757, A155208, A155209, A155210, variants: A155200, A155203.

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

G.f. satisfies: A'(x)/A(x) = 4 + 64*x*A'(16*x)/A(16*x). - Paul D. Hanna, Nov 15 2022

a(n) ~ 4^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A155812 Triangle, read by rows, where g.f.: A(x,y) = exp( Sum_{n>=1} (3^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Original entry on oeis.org

1, 3, 1, 45, 12, 1, 6687, 801, 39, 1, 10782369, 540720, 10764, 120, 1, 169490304819, 3499254081, 29275956, 129348, 363, 1, 25016281429306077, 206071208583660, 709664882337, 1321144632, 1459773, 1092, 1, 34185693516532070487615

Offset: 0

Paul D. Hanna, Feb 04 2009

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,y) = 1 + (3 + y)x + (45 + 12y + y^2)x^2 + (6687 + 801y + 39y^2 + y^3)x^3 +...
Triangle begins:
1;
3, 1;
45, 12, 1;
6687, 801, 39, 1;
10782369, 540720, 10764, 120, 1;
169490304819, 3499254081, 29275956, 129348, 363, 1;
25016281429306077, 206071208583660, 709664882337, 1321144632, 1459773, 1092, 1;
34185693516532070487615, 109444624780070083617, 150302858159634327, 115097787387369, 53628299415, 15815241, 3279, 1; ...

Cf. A155203 (column 0), A155204 (row sums), A155813 (column 1).

{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n+1,(3^m+y)^m*x^m/m)+x*O(x^n)),n,x),k,y)}

G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k.

A155206 G.f.: A(x) = exp( Sum_{n>=1} (3^n - 1)^n/2^(n-1) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 18, 1498, 1283090, 10377556482, 775351592888722, 532444511048570910746, 3349121447720205394546014978, 192371436319107536207473420480152034, 100642626897912335112447860229547933463000450

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} (3^n - 1)^n/2^n * x^n/n ) has fractional coefficients as a power series in x.

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 1498*x^3 + 1283090*x^4 + 10377556482*x^5 +...
log(A(x)) = 2*x + 8^2/2*x^2/2 + 26^3/2^2*x^3/3 + 80^4/2^3*x^4/4 + 242^5/2^4*x^5/5 +...

Cf. A155203, A155204, A155205, A155812 (triangle), variant: A155210.

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

A202630 G.f.: exp( Sum_{n>=1} (3^n + A(x))^n * x^n/n ).

Original entry on oeis.org

1, 4, 62, 7646, 11346032, 173032723944, 25223251091617644, 34295314615208803660344, 429734276354140075492905291038, 49292144933883713910495181570024546094, 51546480948489890934875222750204184228031911158

Offset: 0

Paul D. Hanna, Dec 21 2011

nonn

			G.f.: A(x) = 1 + 4*x + 62*x^2 + 7646*x^3 + 11346032*x^4 + 173032723944*x^5 +...
where
log(A(x)) = (3 + A(x))*x + (3^2 + A(x))^2*x^2/2 + (3^3 + A(x))^3*x^3/3 + (3^4 + A(x))^4*x^4/4 +...

Cf. A202629, A202519, A155203.

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(3^m+A+x*O(x^n))^m*x^m/m)));polcoeff(A,n)}

A156336 G.f.: A(x) = exp( Sum_{n>=1} 3^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 9, 99, 1917, 324567, 65546253, 121237985007, 231991261827633, 4053251131970038227, 71801958531451566872745, 11561440390042361895766055043, 1877401313066393527954697682635421

Offset: 0

Paul D. Hanna, Feb 10 2009

nonn

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 99*x^3 + 1917*x^4 + 324567*x^5 +...
log(A(x)) = 3*x + 3^2*x^2/2 + 3^5*x^3/3 + 3^8*x^4/4 + 3^13*x^5/5 + 3^18*x^6/6 +...

Cf. A156335, A156337, A155203.

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

a(n) = (1/n)*Sum_{k=1..n} 3^floor((k^2+1)/2) * a(n-k) for n>0, with a(0)=1.

A202629 G.f.: exp( Sum_{n>=1} (3^n - A(x))^n * x^n/n ).

Original entry on oeis.org

1, 2, 32, 5872, 10244654, 166008832278, 24810745551644598, 34076373857728228215714, 428687442859626139066325301140, 49247086410581981443124673896698437124, 51529024823944797258322973430879108808780359272

Offset: 0

Paul D. Hanna, Dec 21 2011

nonn

			G.f.: A(x) = 1 + 2*x + 32*x^2 + 5872*x^3 + 10244654*x^4 + 166008832278*x^5 +...
where
log(A(x)) = (3 - A(x))*x + (3^2 - A(x))^2*x^2/2 + (3^3 - A(x))^3*x^3/3 + (3^4 - A(x))^4*x^4/4 +...

Cf. A202630, A202518, A155203.

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(3^m-A+x*O(x^n))^m*x^m/m)));polcoeff(A,n)}

A156911 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2)/(1 - 3^nx)^n x^n/n ).

Original entry on oeis.org

1, 3, 54, 7470, 11326446, 173007630594, 25222890606413004, 34295263336258106333292, 429734207324188407742780371030, 49292144072318945019920850119049478578

Offset: 0

Paul D. Hanna, Feb 17 2009

nonn

An example of this logarithmic identity at q=3:

Sum_{n>=1} [q^(n^2)/(1 - q^n*x)^n]*x^n/n = Sum_{n>=1} [(1 + q^n)^n - 1]*x^n/n.

			G.f.: A(x) = 1 + 3*x + 54*x^2 + 7470*x^3 + 11326446*x^4 +...
Log(A(x)) = 3/(1-3*x)*x + 3^4/(1-3^2*x)^2*x^2/2 + 3^9/(1-3^3*x)^3*x^3/3 +...
Log(A(x)) = (4-1)*x + (10^2-1)*x^2/2 + (28^3-1)*x^3/3 + (82^4-1)*x^4/4 +...

Cf. A155204, A155203, A155205, A156910.

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

/* As First Differences of A155204: */
{a(n)=polcoeff((1-x)*exp(sum(m=1, n+1, (3^m+1)^m*x^m/m)+x*O(x^n)), n)}

G.f.: A(x) = (1-x)*exp( Sum_{n>=1} (1 + 3^n)^n * x^n/n );

Equals the first differences of A155204.

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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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A155204 G.f.: A(x) = exp( Sum_{n>=1} (3^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

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A155205 G.f.: A(x) = exp( Sum_{n>=1} (3^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

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A155207 G.f.: A(x) = exp( Sum_{n>=1} 4^(n^2) * x^n/n ), a power series in x with integer coefficients.

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A155812 Triangle, read by rows, where g.f.: A(x,y) = exp( Sum_{n>=1} (3^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

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A155206 G.f.: A(x) = exp( Sum_{n>=1} (3^n - 1)^n/2^(n-1) * x^n/n ), a power series in x with integer coefficients.

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A202630 G.f.: exp( Sum_{n>=1} (3^n + A(x))^n * x^n/n ).

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A156336 G.f.: A(x) = exp( Sum_{n>=1} 3^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

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A156911 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2)/(1 - 3^nx)^n x^n/n ).