A257673 -id:A257673 - cp's OEIS frontend

A257675 a(n) = A257673(2n,n).

1, 3, 21, 174, 1509, 13473, 122580, 1129999, 10518477, 98644395, 930607321, 8821717743, 83960385396, 801783097911, 7678690148647, 73721697254874, 709323064431597, 6837868454315828, 66028546945097793, 638555320797561440, 6183787002091288969, 59957399899953193063

Offset: 0

Alois P. Heinz, May 03 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000219, A257673.

g:= proc(n) option remember; `if`(n=0, 1, add(
      g(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, g(n+1), (q->
      add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..22);

g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*
     DivisorSigma[2, j], {j, 1, n}]/n];
b[n_, k_] := b[n, k] = If[k < 2, g[n+1], With[{q = Quotient[k, 2]},
     Sum[b[j, q] b[n - j, k - q], {j, 0, n}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

a(n) = A257673(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 9.93288639318036180192949205242384178223421389697248991016311001938239..., c = 0.31807008223273549425589833682845775837952038959... . - Vaclav Kotesovec, May 19 2015
a(n) = [x^(2*n)] (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^n. - Ilya Gutkovskiy, Feb 13 2021

A321947 Column k=2 of triangle A257673.

Original entry on oeis.org

1, 6, 21, 62, 162, 396, 917, 2036, 4380, 9152, 18694, 37380, 73444, 141918, 270370, 508178, 943876, 1733468, 3151396, 5674152, 10126435, 17921016, 31468623, 54848750, 94935565, 163232096, 278903915, 473693432, 799949111, 1343550666, 2244807927, 3731885232

Offset: 2

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Column k=2 of A257673.

Cf. A000219, A161870.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..35);

A321947[n_] := Module[{nn = n}, SeriesCoefficient[Product[1/(1 - x^i)^(2 i), {i, 1, nn}], {x, 0, nn}] - 2*SeriesCoefficient[Product[(1 - x^k)^-k, {k, nn}], {x, 0, nn}]]; Table[A321947[n], {n, 2, 33}] (* Robert P. P. McKone, Jan 30 2021 *)
b[n_, k_] := b[n, k] = If[n == 0, 1, k*Sum[
     b[n - j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
a[n_] := With[{k = 2}, Sum[b[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
Table[a[n], {n, 2, 35}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^2. - Ilya Gutkovskiy, Jan 30 2021

a(n) = A161870(n) - 2*A000219(n). - Vaclav Kotesovec, Jan 30 2021

A321948 Column k=3 of triangle A257673.

Original entry on oeis.org

1, 9, 45, 174, 576, 1719, 4761, 12441, 31050, 74593, 173547, 392787, 867876, 1877322, 3984636, 8314434, 17082510, 34604523, 69194309, 136709688, 267111510, 516515227, 989147760, 1877103486, 3531796959, 6591644601, 12208734552, 22449066710, 40995144288

Offset: 3

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..5000

Column k=3 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
seq(a(n), n=3..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^3. - Ilya Gutkovskiy, Jan 30 2021

A321949 Column k=4 of triangle A257673.

Original entry on oeis.org

1, 12, 78, 376, 1509, 5340, 17234, 51796, 147054, 398388, 1037560, 2612520, 6387965, 15221412, 35446980, 80865304, 181076216, 398660292, 864186408, 1846759404, 3894731430, 8113669352, 16710519860, 34049851236, 68687627812, 137257430140, 271842916654

Offset: 4

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..5000

Column k=4 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(4):
seq(a(n), n=4..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^4. - Ilya Gutkovskiy, Jan 30 2021

A321950 Column k=5 of triangle A257673.

Original entry on oeis.org

1, 15, 120, 695, 3285, 13473, 49730, 169115, 538440, 1623660, 4677121, 12955065, 34682730, 90113220, 227992870, 563267203, 1361992935, 3229643480, 7522847555, 17237982025, 38905739524, 86585024910, 190193593830, 412712252535, 885382820550, 1879084411753

Offset: 5

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..5000

Column k=5 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):
seq(a(n), n=5..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^5. - Ilya Gutkovskiy, Jan 30 2021

A321951 Column k=6 of triangle A257673.

Original entry on oeis.org

1, 18, 171, 1158, 6309, 29466, 122580, 465738, 1644516, 5464892, 17253369, 52128540, 151592391, 426265836, 1163373243, 3091338000, 8018585046, 20348618814, 50615278427, 123608650182, 296794147017, 701525018576, 1634144413185, 3755097200094, 8519488746222

Offset: 6

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..5000

Column k=6 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(6):
seq(a(n), n=6..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^6. - Ilya Gutkovskiy, Jan 30 2021

A321952 Column k=7 of triangle A257673.

Original entry on oeis.org

1, 21, 231, 1792, 11067, 58044, 268940, 1129999, 4385136, 15928948, 54711958, 179090772, 562156203, 1700628930, 4978677738, 14153099499, 39180254316, 105881154624, 279906223856, 725158329175, 1844006226090, 4608929551309, 11336379967178, 27469729015029

Offset: 7

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..5000

Column k=7 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(7):
seq(a(n), n=7..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^7. - Ilya Gutkovskiy, Jan 30 2021

A321953 Column k=8 of triangle A257673.

Original entry on oeis.org

1, 24, 300, 2624, 18126, 105552, 539408, 2485016, 10518477, 41482336, 154055260, 543239064, 1830924554, 5929728456, 18534968236, 56121729792, 165117049094, 473276306552, 1324582728412, 3626879184272, 9732325392280, 25631811881168, 66342981204768

Offset: 8

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..5000

Column k=8 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(8):
seq(a(n), n=8..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^8. - Ilya Gutkovskiy, Jan 31 2021

A321954 Column k=9 of triangle A257673.

Original entry on oeis.org

1, 27, 378, 3681, 28134, 180198, 1007370, 5051790, 23173047, 98644395, 394006761, 1489460724, 5365964511, 18526724685, 61587671283, 197885754837, 616568620176, 1868127089697, 5517376711191, 15917134652829, 44935916034951, 124340811582540, 337706753653476

Offset: 9

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..5000

Column k=9 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(9):
seq(a(n), n=9..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^9. - Ilya Gutkovskiy, Jan 31 2021

A321955 Column k=10 of triangle A257673.

Original entry on oeis.org

1, 30, 465, 4990, 41820, 292296, 1775075, 9629800, 47604225, 217630430, 930607321, 3755443890, 14405010340, 52827550470, 186123730845, 632552752322, 2080824994210, 6644958372540, 20652164516930, 62605166996300, 185464736482827, 537841680016510, 1529116657680575

Offset: 10

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..5000

Column k=10 of A257673.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(10):
seq(a(n), n=10..35);

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^10. - Ilya Gutkovskiy, Jan 31 2021

cp's OEIS Frontend

A257675 a(n) = A257673(2n,n).

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A321947 Column k=2 of triangle A257673.

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A321948 Column k=3 of triangle A257673.

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A321949 Column k=4 of triangle A257673.

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A321950 Column k=5 of triangle A257673.

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A321951 Column k=6 of triangle A257673.

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A321952 Column k=7 of triangle A257673.

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A321953 Column k=8 of triangle A257673.

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A321954 Column k=9 of triangle A257673.

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