A023882 -id:A023882 - cp's OEIS frontend

A023881 Number of partitions in expanding space: sigma(n,q) is the sum of the q-th powers of the divisors of n.

1, 1, 3, 12, 82, 725, 8811, 128340, 2257687, 45658174, 1052672116, 27108596725, 772945749970, 24137251258926, 819742344728692, 30069017799172228, 1184889562926838573, 49914141857616862435

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 3*x^2 + 12*x^3 + 82*x^4 + 725*x^5 + 8811*x^6 + 128340*x^7 + 2257687*x^8 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Cf. A023882, A023887, A158952.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-k^k*x^k)^(1/k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

seq(coeff(series(mul((1-k^k*x^k)^(-1/k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

nmax=30; CoefficientList[Series[Product[1/(1-k^k*x^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 31 2018 *)

{a(n) = if( n<0, 0, polcoeff( exp( sum( k=1, n, sigma(k, k) * x^k / k, x * O(x^n))), n))} /* Michael Somos, Feb 15 2006 */

{a(n)=if(n<0,0,polcoeff(prod(k=1,n,(1-k^k*x^k+x*O(x^n))^(-1/k)),n))} /* Paul D. Hanna */

G.f.: exp( Sum_{k>0} sigma_k(k) * x^k / k). - Michael Somos, Feb 15 2006
G.f.: Product_{n>=1} (1 - n^n*x^n)^(-1/n). - Paul D. Hanna, Mar 08 2011
a(n) ~ n^(n-1). - Vaclav Kotesovec, Oct 08 2016

A294645 a(n) = Sum_{d|n} d^(n+1).

Original entry on oeis.org

1, 9, 82, 1057, 15626, 282252, 5764802, 134480385, 3486843451, 100048830174, 3138428376722, 107006334784468, 3937376385699290, 155572843119354936, 6568408508343827972, 295150156996346511361, 14063084452067724991010, 708236696816416252145973

Offset: 1

Seiichi Manyama, Nov 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Divisor Function

Column k=1 of A308504.

Cf. A023882, A023887, A082245, A158095, A292312.

Table[DivisorSigma[n + 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Oct 07 2020 *)

PARI
```
{a(n) = sigma(n, n+1)}
```

N=66; x='x+O('x^N); Vec(sum(k=1, N, k^(k+1)*x^k/(1-(k*x)^k)))

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-(k*x)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>0} k^(k+1)*x^k/(1-(k*x)^k).
L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 02 2019
a(n) ~ n^(n+1). - Vaclav Kotesovec, Oct 07 2020

A265949 Expansion of Product_{k>=1} (1 + k^k*x^k).

Original entry on oeis.org

1, 1, 4, 31, 283, 3489, 50913, 890635, 17891170, 409850236, 10494427982, 297780829216, 9261266862273, 313453533534739, 11464487066049791, 450644378868285130, 18942868694407904729, 847930346323808122469, 40266107916200371331007, 2021842180288047801103956

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A023882, A292190, A292305, A292306.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

seq(coeff(series(mul((1+k^k*x^k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

nmax=20; CoefficientList[Series[Product[(1+k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

m=30; x='x+O('x^m); Vec(prod(k=1, m, (1+k^k*x^k))) \\ G. C. Greubel, Oct 31 2018

a(n) ~ n^n * (1 + exp(-1)/n + ((1/2)*exp(-1) + 4*exp(-2))/n^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d + 1)*d^(k+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018

A292312 Expansion of Product_{k>=1} (1 - k^k*x^k).

Original entry on oeis.org

1, -1, -4, -23, -229, -2761, -42615, -758499, -15702086, -365588036, -9516954786, -273061566624, -8575969258607, -292418459301779, -10762887030763337, -425243370397722674, -17953905924215881215, -806666656048846472309

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Column k=1 of A294653.

Cf. A023882, A265949.

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 - k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

seq(coeff(series(mul((1-k^k*x^k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

terms = 18; CoefficientList[Product[(1 - k^k*x^k), {k, 1, terms}] + O[x]^(terms), x] (* Jean-François Alcover, Nov 11 2017 *)

{a(n) = polcoeff(prod(k=1, n, 1-k^k*x^k+x*O(x^n)), n)}

Convolution inverse of A023882.
a(n) ~ -n^n * (1 - exp(-1)/n - (exp(-1)/2 + 4*exp(-2))/n^2). - Vaclav Kotesovec, Sep 14 2017
a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294645(k)*a(n-k) for n > 0. - Seiichi Manyama, Nov 09 2017

A356530 Expansion of e.g.f. Product_{k>0} 1/(1 - (k * x)^k)^(1/k^k).

Original entry on oeis.org

1, 1, 4, 18, 156, 1020, 23040, 189000, 8462160, 174741840, 8418513600, 110288455200, 26670240273600, 364684824504000, 46300470369753600, 5169242034644688000, 359472799348030368000, 7508907247291081632000, 6157317530690533823616000

Offset: 0

Seiichi Manyama, Aug 10 2022

nonn

Cf. A023882, A294462, A356487, A356529.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k^k))))

a356529(n) = (n-1)!*sumdiv(n, d, d^(n-d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356529(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A356529(k) * binomial(n-1,k-1) * a(n-k).

A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 31, 57, 113, 212, 410, 757, 1464, 2684, 5083, 9380, 17569, 32120, 59977, 109193, 202046, 367951, 675541, 1224453, 2243795, 4052369, 7377243, 13314989, 24140198, 43406515, 78510429, 140800279, 253663615, 454352111, 815790813, 1457485309

Offset: 0

Vaclav Kotesovec, Mar 08 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500

Cf. A000045, A023882, A075900, A077365, A179381, A318248.

nmax = 40; CoefficientList[Series[Product[1/(1-Fibonacci[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ log(phi)*n + 2*sqrt(polylog(2, 1/sqrt(5))*n) - 3*(log(n)/4), where polylog(2, 1/sqrt(5)) = 0.5107013915606224266804289751265205446721... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio.

A294758 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 32, 5, 1, 1, 65, 746, 304, 7, 1, 1, 257, 19748, 66538, 3537, 11, 1, 1, 1025, 531698, 16801060, 9843827, 52010, 15, 1, 1, 4097, 14349932, 4295564530, 30535638897, 2188210276, 895397, 22, 1, 1, 16385, 387424586, 1099527026284, 95371863254051, 101591953731770, 680615495493, 18016416, 30

Offset: 0

Seiichi Manyama, Nov 08 2017

			Square array begins:
   1,   1,     1,        1, ...
   1,   1,     1,        1, ...
   2,   5,    17,       65, ...
   3,  32,   746,    19748, ...
   5, 304, 66538, 16801060, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..1 give A000041, A023882.
Rows n=0-1 give A000012.
Cf. A294653.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j)) * A(n-j,k) for n > 0.

A356487 Expansion of e.g.f. Product_{k>0} 1/(1 - (k * x)^k)^(1/k!).

Original entry on oeis.org

1, 1, 6, 45, 580, 7105, 170076, 2654575, 116426528, 2386183761, 209503380160, 3455683548691, 969334978024920, 15164681616944353, 6510178188269825720, 223847763757748796975, 81261936394687862700256, 1581790511799886415713825

Offset: 0

Seiichi Manyama, Aug 09 2022

nonn

Cf. A023882, A209902, A356486.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k!))))

a356486(n) = (n-1)!*sumdiv(n, d, d^n/(d-1)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356486(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A356486(k) * binomial(n-1,k-1) * a(n-k).

A292311 Expansion of Product_{k>=1} 1/(1 + k^k*x^k).

Original entry on oeis.org

1, -1, -3, -24, -216, -2801, -42166, -762397, -15685040, -366477168, -9523974486, -273453483050, -8583651341879, -292700900034984, -10770969729108326, -425541512224476567, -17964544188354355022, -807097409926675847400

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A023882, A265949.

{a(n) = polcoeff(1/prod(k=1, n, 1+k^k*x^k+x*O(x^n)), n)}

Convolution inverse of A265949.

a(n) ~ -n^n * (1 - exp(-1)/n - (exp(-1)/2 + 3*exp(-2))/n^2). - Vaclav Kotesovec, Sep 14 2017

A292406 Expansion of Product_{k>=1} ((1 + k^kx^k)/(1 - k^kx^k)).

Original entry on oeis.org

1, 2, 10, 72, 670, 7896, 113572, 1939028, 38463550, 869985586, 22098989952, 622728621984, 19271496576612, 649553583740576, 23680212403186584, 928276782505698920, 38931911577966732814, 1739307919812511213916, 82457732209611432170734

Offset: 0

Seiichi Manyama, Sep 15 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A023882, A265949, A292407.

nmax = 20; CoefficientList[Series[Product[(1 + k^k*x^k)/(1 - k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2017 *)

Convolution of A023882 and A265949.

a(n) ~ 2*n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - Vaclav Kotesovec, Sep 16 2017

cp's OEIS Frontend

A023881 Number of partitions in expanding space: sigma(n,q) is the sum of the q-th powers of the divisors of n.

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A294645 a(n) = Sum_{d|n} d^(n+1).

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PARI

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A265949 Expansion of Product_{k>=1} (1 + k^k*x^k).

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Magma

Maple

Mathematica

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A292312 Expansion of Product_{k>=1} (1 - k^k*x^k).

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Magma

Maple

Mathematica

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A356530 Expansion of e.g.f. Product_{k>0} 1/(1 - (k * x)^k)^(1/k^k).

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PARI

PARI

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A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

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Mathematica

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A294758 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(k*j)*x^j) in powers of x.

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A356487 Expansion of e.g.f. Product_{k>0} 1/(1 - (k * x)^k)^(1/k!).

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A294758 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j) in powers of x.

A292406 Expansion of Product_{k>=1} ((1 + k^kx^k)/(1 - k^kx^k)).