A343349 -id:A343349 - cp's OEIS frontend

A343350 Expansion of Product_{k>=1} 1 / (1 - x^k)^(5^(k-1)).

1, 1, 6, 31, 171, 921, 5031, 27281, 148101, 801901, 4336902, 23415777, 126254962, 679805112, 3655679442, 19634501447, 105334380517, 564471596667, 3021754455157, 16160029793032, 86339725851558, 460874548444683, 2457961986888773, 13097958657023523, 69740119667456018

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144069, A343349, A343351, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*5^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 12 2021

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

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 * (5^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343351 Expansion of Product_{k>=1} 1 / (1 - x^k)^(6^(k-1)).

Original entry on oeis.org

1, 1, 7, 43, 280, 1792, 11586, 74550, 479892, 3083640, 19794678, 126908502, 812761299, 5199586119, 33230586285, 212172173565, 1353444677529, 8626044781761, 54931168743703, 349524243121795, 2222294161109422, 14119034725444774, 89639674321304392, 568720801952770012

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144070, A343349, A343350, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*6^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

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

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 * (6^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343352 Expansion of Product_{k>=1} 1 / (1 - x^k)^(7^(k-1)).

Original entry on oeis.org

1, 1, 8, 57, 428, 3172, 23689, 176324, 1312550, 9757798, 72480269, 537854094, 3987751860, 29540543908, 218652961074, 1617159619805, 11951595353413, 88264810625245, 651404299886762, 4804261815210433, 35410065096578748, 260832137791524693, 1920169120639498017, 14127684273966098698

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144071, A343349, A343350, A343351, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*7^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

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

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 * (7^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343353 Expansion of Product_{k>=1} 1 / (1 - x^k)^(8^(k-1)).

Original entry on oeis.org

1, 1, 9, 73, 621, 5229, 44293, 374277, 3162447, 26694159, 225163687, 1897751079, 15983278059, 134519816427, 1131395821587, 9509592524371, 79880259426102, 670590654977718, 5626336598011078, 47179486350900358, 395410837699366686, 3312225325409475038, 27731588831310844302

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144072, A343349, A343350, A343351, A343352, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*8^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

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

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 * (8^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343354 Expansion of Product_{k>=1} 1 / (1 - x^k)^(9^(k-1)).

Original entry on oeis.org

1, 1, 10, 91, 865, 8155, 77251, 730435, 6905560, 65233120, 615847378, 5810270782, 54784324495, 516250199827, 4862041512625, 45765734635702, 430560567351208, 4048630897384450, 38051334554031551, 357459295903931045, 3356488167698692226, 31503001136703776561

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144073, A343349, A343350, A343351, A343352, A343353, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*9^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

nmax = 21; CoefficientList[Series[Product[1/(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[d 9^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

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 * (9^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343355 Expansion of Product_{k>=1} 1 / (1 - x^k)^(10^(k-1)).

Original entry on oeis.org

1, 1, 11, 111, 1166, 12166, 127436, 1332936, 13939651, 145683351, 1521743103, 15886781603, 165770328383, 1728861822083, 18022063489023, 187778810866043, 1955660195168328, 20358764860253028, 211849198103034998, 2203562708619192998, 22911457758236641451, 238129937419462634151

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

In general, if m > 1 and g.f. = Product_{k>=1} 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 * (m^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

Cf. A034691, A104460, A292837, A343349, A343350, A343351, A343352, A343353, A343354.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*10^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

nmax = 21; CoefficientList[Series[Product[1/(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[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 * (10^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 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

cp's OEIS Frontend

A343350 Expansion of Product_{k>=1} 1 / (1 - x^k)^(5^(k-1)).

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A343351 Expansion of Product_{k>=1} 1 / (1 - x^k)^(6^(k-1)).

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A343352 Expansion of Product_{k>=1} 1 / (1 - x^k)^(7^(k-1)).

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A343353 Expansion of Product_{k>=1} 1 / (1 - x^k)^(8^(k-1)).

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A343354 Expansion of Product_{k>=1} 1 / (1 - x^k)^(9^(k-1)).

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A343355 Expansion of Product_{k>=1} 1 / (1 - x^k)^(10^(k-1)).

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

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