A265951 -id:A265951 - cp's OEIS frontend

A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

1, 3, 15, 54, 210, 699, 2484, 7995, 26610, 84186, 269940, 839238, 2634579, 8098194, 25032282, 76388265, 233791104, 709501596, 2157488730, 6523204836, 19747491810, 59558682132, 179762506329, 541222906812, 1630300772106, 4902697929306, 14748249476553

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006906, A265951, A265975, A265976.

Cf. A246935, A242587.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      3^n, b(n, i-1) +i*3*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-3*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^n, where c = Product_{m>=2} 1/(1 - m/3^(m-1)) = 5.86277744540963226378877460838259757442241952947887939654316926419876...

A265975 Expansion of Product_{k>=1} 1/(1 - 4kx^k).

Original entry on oeis.org

1, 4, 24, 108, 512, 2164, 9464, 39004, 163008, 663588, 2713752, 10954764, 44328512, 178160724, 716821752, 2874497660, 11532111232, 46187508676, 185028540696, 740595436652, 2964628293504, 11862432443764, 47467812675320, 189902835709212, 759756868215872

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006906, A265951, A265974, A265976.

Cf. A246935, A246936.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      4^n, b(n, i-1) +i*4*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-4*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = Product_{m>=2} 1/(1 - m/4^(m-1)) = 2.700170514502619666262858845683166558216386190684736249639219328278569...

A265976 Expansion of Product_{k>=1} 1/(1 - 5kx^k).

Original entry on oeis.org

1, 5, 35, 190, 1070, 5525, 29080, 147485, 752790, 3789170, 19105800, 95794930, 480650335, 2406018490, 12047084370, 60264282575, 301493182380, 1507758356660, 7540528037090, 37705593514220, 188545393000350, 942756783659980, 4713958620697385, 23570092258449540

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006906, A265951, A265974, A265975.

Cf. A246935, A246937.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      5^n, b(n, i-1) +i*5*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-5*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 5^n, where c = Product_{m>=2} 1/(1 - m/5^(m-1)) = 1.977268427518901757865749340705853730491796767544158844539130847296...

A032309 "EFK" (unordered, size, unlabeled) transform of 2,4,6,8,...

Original entry on oeis.org

1, 2, 4, 14, 20, 50, 112, 190, 328, 666, 1340, 2038, 3740, 5954, 10792, 19542, 30048, 48290, 80164, 124694, 204484, 347610, 515184, 810750, 1240296, 1932722, 2887820, 4557838, 7126652, 10463330, 15768168, 23499934

Offset: 0

Christian G. Bower

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
C. G. Bower, Transforms (2)

Cf. A022629, A032302, A265951, A265955.

nmax=40; CoefficientList[Series[Product[(1+2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 19 2015 *)

seq(n)={Vec(prod(k=1, n, 1 + 2*k*x^k + O(x*x^n)))} \\ Andrew Howroyd, Sep 20 2018

G.f.: Product_{k >= 1} (1 + 2*k*x^k).

a(0)=1 prepended by Andrew Howroyd, Sep 20 2018

A265955 Expansion of Product_{k>=1} (1 + 2kx^k)/(1 - 2kx^k).

Original entry on oeis.org

1, 4, 16, 60, 192, 596, 1776, 5020, 13760, 36916, 96336, 246316, 619392, 1530548, 3729392, 8976364, 21337920, 50195268, 116977232, 270114764, 618712640, 1406843940, 3176387120, 7126185948, 15894370816, 35253947940, 77796242768, 170868178332, 373606888128

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3280

Cf. A006906, A022629, A032302, A032309, A070933, A265951.

nmax=40; CoefficientList[Series[Product[(1+2*k*x^k)/(1-2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * n * 2^n, where c = 2 * Product_{m>=3} (1 + 2/(2^(m-1)/m - 1)) = 193.4198278838721371054040810054045645734538119720773785523616944906739...

A266137 Expansion of Product_{k>=1} 1/(1 - k(x^(2k-1))).

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 14, 20, 33, 50, 68, 106, 147, 214, 325, 445, 624, 916, 1259, 1780, 2553, 3477, 4821, 6794, 9340, 12777, 17808, 24266, 32998, 45764, 61770, 83593, 114594, 154039, 208617, 283232, 379040, 509270, 687448, 919709, 1228319, 1650595, 2195745

Offset: 0

Vaclav Kotesovec, Dec 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..9850

Cf. A000009, A006906, A077335, A265951, A266138.

nmax=60; CoefficientList[Series[Product[1/(1-k*(x^(2*k-1))), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 2^(n/3), where
c = 2684.3207660224428945778151546260301591494083790... if mod(n,3) = 0
c = 2683.9203893332021512699407898064547843826991184... if mod(n,3) = 1
c = 2683.7635451650373491773203224442103370428384569... if mod(n,3) = 2.

A266138 Expansion of Product_{k>=1} 1/(1 - k(x^(2k+1))).

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 7, 7, 11, 13, 24, 26, 35, 44, 69, 78, 112, 150, 188, 245, 318, 429, 537, 729, 924, 1177, 1534, 1965, 2518, 3287, 4108, 5394, 6857, 8604, 11022, 14073, 17899, 22549, 28900, 36182, 45954, 58395, 72912, 92118, 116201, 146279

Offset: 0

Vaclav Kotesovec, Dec 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006906, A077335, A087897, A265951, A266137.

nmax=80; CoefficientList[Series[Product[1/(1-k*(x^(2*k+1))), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(n/7), where
c = 617630.638335... if mod(n,7) = 0
c = 617630.321433... if mod(n,7) = 1
c = 617630.360795... if mod(n,7) = 2
c = 617630.429073... if mod(n,7) = 3
c = 617630.357078... if mod(n,7) = 4
c = 617630.421636... if mod(n,7) = 5
c = 617630.341606... if mod(n,7) = 6.

cp's OEIS Frontend

A265974 Expansion of Product_{k>=1} 1/(1 - 3*k*x^k).

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A265975 Expansion of Product_{k>=1} 1/(1 - 4*k*x^k).

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A265976 Expansion of Product_{k>=1} 1/(1 - 5*k*x^k).

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A032309 "EFK" (unordered, size, unlabeled) transform of 2,4,6,8,...

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A265955 Expansion of Product_{k>=1} (1 + 2*k*x^k)/(1 - 2*k*x^k).

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A266137 Expansion of Product_{k>=1} 1/(1 - k*(x^(2*k-1))).

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A266138 Expansion of Product_{k>=1} 1/(1 - k*(x^(2*k+1))).

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A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

A265975 Expansion of Product_{k>=1} 1/(1 - 4kx^k).

A265976 Expansion of Product_{k>=1} 1/(1 - 5kx^k).

A265955 Expansion of Product_{k>=1} (1 + 2kx^k)/(1 - 2kx^k).

A266137 Expansion of Product_{k>=1} 1/(1 - k(x^(2k-1))).

A266138 Expansion of Product_{k>=1} 1/(1 - k(x^(2k+1))).