A343360 -id:A343360 - cp's OEIS frontend

A343363 Expansion of Product_{k>=1} (1 + x^k)^(6^(k-1)).

1, 1, 6, 42, 267, 1743, 11234, 72470, 466251, 2996883, 19234836, 123315828, 789682546, 5051601010, 32282443044, 206104519572, 1314652656453, 8378283675645, 53350205335626, 339445117302366, 2158091256282273, 13710402587540469, 87040883294333382, 552205562345916570

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A098407, A292840, A343351, A343360, A343361, A343362, A343364, A343365, A343366.

N:= 100: # for a(0)..a(N)
G:= mul((1+x^k)^(6^(k-1)),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Apr 12 2021

nmax = 23; CoefficientList[Series[Product[(1 + x^k)^(6^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 6^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(6^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(sqrt(2*n/3) - 1/12 - c/6) * 6^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (6^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A343361 Expansion of Product_{k>=1} (1 + x^k)^(4^(k-1)).

Original entry on oeis.org

1, 1, 4, 20, 86, 390, 1724, 7644, 33697, 148401, 651584, 2855840, 12491276, 54540636, 237733768, 1034610232, 4495832776, 19508749928, 84540638312, 365888222552, 1581630245756, 6829047398156, 29453496620000, 126898489491904, 546183557447366, 2348560270762006, 10089340886428928

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A098407, A292838, A343349, A343360, A343362, A343363, A343364, A343365, A343366.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(4^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 12 2021

nmax = 26; CoefficientList[Series[Product[(1 + x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(4^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(sqrt(n) - 1/8 - c/4) * 2^(2*n - 3/2) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (4^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A343362 Expansion of Product_{k>=1} (1 + x^k)^(5^(k-1)).

Original entry on oeis.org

1, 1, 5, 30, 160, 885, 4810, 26185, 142005, 769305, 4159301, 22455876, 121057525, 651737675, 3504241650, 18818709130, 100945053055, 540885242825, 2895159035375, 15481318817450, 82704855762375, 441427664993275, 2354020475714775, 12542918682786300, 66778882780674975

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A098407, A292839, A343350, A343360, A343361, A343363, A343364, A343365, A343366.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(5^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 12 2021

nmax = 24; CoefficientList[Series[Product[(1 + x^k)^(5^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 5^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(5^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(2*sqrt(n/5) - 1/10 - c/5) * 5^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (5^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A343364 Expansion of Product_{k>=1} (1 + x^k)^(7^(k-1)).

Original entry on oeis.org

1, 1, 7, 56, 413, 3108, 23163, 172711, 1285256, 9556603, 70980000, 526711507, 3904946864, 28926003505, 214095348671, 1583389916081, 11701578676851, 86415267247743, 637732279701496, 4703270177738076, 34664585073280204, 255332979654402524, 1879629724498860397, 13829015594546304600

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A098407, A292841, A343352, A343360, A343361, A343362, A343363, A343365, A343366.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(7^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

nmax = 23; CoefficientList[Series[Product[(1 + x^k)^(7^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 7^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(7^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(2*sqrt(n/7) - 1/14 - c/7) * 7^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (7^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A343365 Expansion of Product_{k>=1} (1 + x^k)^(8^(k-1)).

Original entry on oeis.org

1, 1, 8, 72, 604, 5148, 43544, 368408, 3112262, 26273542, 221605240, 1867736120, 15730022540, 132385106956, 1113413229000, 9358220560136, 78606905495809, 659886123312449, 5536404584185376, 46424396382193376, 389074608184431328, 3259085506224931424, 27286163457927575200

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A098407, A292842, A343353, A343360, A343361, A343362, A343363, A343364, A343366.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(8^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(8^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 8^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(8^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(sqrt(n/2) - 1/16 - c/8) * 2^(3*n - 7/4) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (8^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A343366 Expansion of Product_{k>=1} (1 + x^k)^(9^(k-1)).

Original entry on oeis.org

1, 1, 9, 90, 846, 8055, 76224, 721389, 6819192, 64422126, 608173020, 5737815756, 54100140735, 509794737636, 4801164836634, 45192001954005, 425156458320783, 3997756503852489, 37572655020653089, 352957677187938076, 3314174696310855888, 31105460092251410001, 291818245344169918725

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)^(m^(k-1)), then a(n, m) ~ exp(2*sqrt(n/m) - 1/(2*m) - c(m)/m) * m^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c(m) = Sum_{j>=2} (-1)^j / (j * (m^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

Cf. A098407, A292843, A343354, A343360, A343361, A343362, A343363, A343364, A343365.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(9^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(9^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 9^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(9^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(2*sqrt(n/9) - 1/18 - c/9) * 9^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (9^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A343331 Expansion of Product_{k>=1} (1 + x^k)^(10^(k-1)).

Original entry on oeis.org

1, 1, 10, 110, 1145, 12045, 126070, 1319570, 13798710, 144217910, 1506406702, 15726571002, 164096557935, 1711386871635, 17839701265570, 185876723016390, 1935830424374840, 20152131324766520, 209696974024339610, 2181155691766631710, 22678274833738085501, 235704268837407670401

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)^(m^(k-1)), then a(n, m) ~ exp(2*sqrt(n/m) - 1/(2*m) - c(m)/m) * m^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c(m) = Sum_{j>=2} (-1)^j / (j * (m^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

Cf. A098407, A292844, A343355, A343360, A343361, A343362, A343363, A343364, A343365, A343366.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(10^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(10^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 10^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

a(n) ~ exp(sqrt(2*n/5) - 1/20 - c/10) * 10^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (10^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

cp's OEIS Frontend

A343363 Expansion of Product_{k>=1} (1 + x^k)^(6^(k-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A343361 Expansion of Product_{k>=1} (1 + x^k)^(4^(k-1)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A343362 Expansion of Product_{k>=1} (1 + x^k)^(5^(k-1)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A343364 Expansion of Product_{k>=1} (1 + x^k)^(7^(k-1)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A343365 Expansion of Product_{k>=1} (1 + x^k)^(8^(k-1)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A343366 Expansion of Product_{k>=1} (1 + x^k)^(9^(k-1)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A343331 Expansion of Product_{k>=1} (1 + x^k)^(10^(k-1)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula