A343355 -id:A343355 - cp's OEIS frontend

A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

1, 1, 5, 21, 95, 415, 1851, 8155, 36030, 158510, 696502, 3052966, 13359230, 58346206, 254405630, 1107479694, 4813850699, 20894227355, 90567536543, 392066476815, 1695180397145, 7320927664713, 31581573600685, 136094434672509, 585876330191950, 2519701493092958

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144068, A343350, A343351, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*4^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 12 2021

nmax = 25; CoefficientList[Series[Product[1/(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[d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

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

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

Original entry on oeis.org

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

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

A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

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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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Maple

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

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Maple

Mathematica

Formula

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

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