cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

Original entry on oeis.org

1, -2, 2, -6, 14, -26, 50, -102, 214, -426, 834, -1678, 3398, -6778, 13482, -27022, 54198, -108306, 216346, -432878, 866334, -1732386, 3463626, -6927926, 13858350, -27715378, 55426002, -110855030, 221719582, -443433610, 886848930, -1773709078, 3547455846
Offset: 0

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 27 2002

Keywords

Links

Crossrefs

Cf. A070933, A032302, A000700, A246935, A261562, A261566, A261582.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(O[x]^30 + 3/QPochhammer[-2, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1-1/(-2)^j) = 1/QPochhammer[-1/2,-1/2] = 0.8259519860658427384636116224100201356301... . - Vaclav Kotesovec, Aug 25 2015
G.f.: Sum_{i>=0} (-2)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

Extensions

More terms from Vaclav Kotesovec, Aug 25 2015

A279360 Expansion of Product_{k>=1} (1+2*x^(k^2)).

Original entry on oeis.org

1, 2, 0, 0, 2, 4, 0, 0, 0, 2, 4, 0, 0, 4, 8, 0, 2, 4, 0, 0, 4, 8, 0, 0, 0, 6, 12, 0, 0, 12, 24, 0, 0, 0, 4, 8, 2, 4, 8, 16, 4, 12, 8, 0, 0, 12, 24, 0, 0, 10, 28, 16, 4, 12, 24, 32, 8, 16, 4, 8, 0, 12, 32, 16, 2, 32, 56, 0, 4, 16, 24, 16, 0, 4, 36, 56, 0, 16
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2016

Keywords

Links

Crossrefs

Cf. A001422, A032302, A033461, A279226, A279368, A341040.

Programs

  • Mathematica
    nmax = 200; CoefficientList[Series[Product[(1+2*x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 0; Do[Do[poly[[j + 1]] += 2*poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; Take[poly, nmax+1]

Formula

a(n) ~ c^(1/3) * exp(3 * 2^(-4/3) * c^(2/3) * Pi^(1/3) * n^(1/3)) / (3 * 2^(2/3) * Pi^(1/3) * n^(5/6)), where c = -PolyLog(3/2, -2) = 1.28138038315976963883198... . - Vaclav Kotesovec, Dec 12 2016
From Alois P. Heinz, Feb 03 2021: (Start)
a(n) = Sum_{k>=0} 2^k * A341040(n,k).
a(n) = 0 <=> n in { A001422 }. (End)

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

Views

Author

Christian G. Bower

Keywords

Links

Crossrefs

Cf. A022629, A032302, A265951, A265955.

Programs

  • Mathematica
    nmax=40; CoefficientList[Series[Product[(1+2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 19 2015 *)
  • PARI
    seq(n)={Vec(prod(k=1, n, 1 + 2*k*x^k + O(x*x^n)))} \\ Andrew Howroyd, Sep 20 2018

Formula

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

Extensions

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

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

Views

Author

Vaclav Kotesovec, Dec 19 2015

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

A266819 Expansion of Product_{k>=1} ((1 + x^k) * (1 + 2*x^k)).

Original entry on oeis.org

1, 3, 5, 12, 20, 33, 60, 93, 144, 222, 340, 498, 729, 1050, 1486, 2115, 2946, 4068, 5592, 7608, 10278, 13854, 18483, 24528, 32426, 42594, 55677, 72498, 94008, 121290, 156002, 199842, 255012, 324438, 411318, 519771, 655128, 823056, 1031148, 1288590, 1605945
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 04 2016

Keywords

Comments

Convolution of A000009 and A032302.

Links

Crossrefs

Cf. A000009, A032302, A266820, A266822.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/4 + log(2)^2/2 + polylog(2, -1/2) = 2.259213400307794164599109607216595948859... .

A266820 Expansion of Product_{k>=1} ((1 + 2*x^k) * (1 + 3*x^k)).

Original entry on oeis.org

1, 5, 11, 30, 66, 115, 252, 445, 762, 1350, 2238, 3690, 5909, 9480, 14460, 22475, 34326, 51150, 76398, 111810, 163350, 236610, 339667, 482040, 684060, 960780, 1340953, 1863570, 2573022, 3533310, 4830822, 6580170, 8900382, 12011430, 16125198, 21567965
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 04 2016

Keywords

Comments

Convolution of A032302 and A032308.
In general, for m1 > 0 and m2 > 0, if g.f. = Product_{k>=1} ((1 + m1*x^k) * (1 + m2*x^k)) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m1+1)*(m2+1)*Pi) * n^(3/4)), where c = Pi^2/3 + log(m1)^2/2 + log(m2)^2/2 + polylog(2, -1/m1) + polylog(2, -1/m2).

Links

Crossrefs

Cf. A032302, A032308, A266819, A266822.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1+2*x^k) * (1+3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi) * n^(3/4)), where c = Pi^2/3 + log(2)^2/2 + log(3)^2/2 + polylog(2, -1/2) + polylog(2, -1/3) = 6.665989921346842772385004076363525173910446415877... .

A291970 Triangle read by rows: T(n,k) = 2 * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 2, 0, 2, 0, 2, 4, 0, 2, 4, 0, 2, 8, 0, 2, 8, 8, 0, 2, 12, 8, 0, 2, 12, 16, 0, 2, 16, 24, 0, 2, 16, 32, 16, 0, 2, 20, 40, 16, 0, 2, 20, 56, 32, 0, 2, 24, 64, 48, 0, 2, 24, 80, 80, 0, 2, 28, 96, 96, 32, 0, 2, 28, 112, 144, 32, 0, 2, 32, 128, 176, 64, 0, 2, 32
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2017

Keywords

Examples

			First few rows are:
  1;
  0, 2;
  0, 2;
  0, 2,  4;
  0, 2,  4;
  0, 2,  8;
  0, 2,  8,  8;
  0, 2, 12,  8;
  0, 2, 12, 16;
  0, 2, 16, 24;
  0, 2, 16, 32, 16.

Links

Crossrefs

Row sums give A032302.
Columns 0-1 give A000007, A007395.
Cf. A008289 (m=1), this sequence (m=2), A291971 (m=3).

A370736 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/4).

Original entry on oeis.org

1, 2, 2, 76, -106, 1788, -1516, 57176, -276634, 2270444, -10094212, 97699752, -664173444, 4819718488, -33236872088, 259931360688, -1894783205754, 13983087008588, -103270227527444, 779496572387208, -5855545477963244, 44016069418771976, -331519650617078376, 2514477954420678352
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

Cf. A032302 (m=1), A370709 (m=2), A370716 (m=3), A370737 (m=5).

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[1+2*x^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[1+2*(4*x)^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + 2*(4*x)^k)^(1/4).
a(n) ~ (-1)^(n+1) * QPochhammer(-1/2)^(1/4) * 8^n / (4 * Gamma(3/4) * n^(5/4)).

A370739 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, -75, 35250, -1138125, 72645000, -3307996875, 244578890625, -15502648125000, 985908809765625, -63515254624218750, 4314500023927734375, -291905297026816406250, 19789483493484814453125, -1355414138248614990234375, 93666904586649390380859375, -6498800175020013123779296875
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Comments

In general, if d > 1, m > 1 and g.f. = Product_{k>=1} (1 + d*x^k)^(1/m), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d)^(1/m) * d^n / (m*Gamma(1 - 1/m) * n^(1 + 1/m)).

Crossrefs

Cf. A032308 (d=3,m=1), A370711 (d=3,m=2), A370712 (d=3,m=3), A370738 (d=3,m=4).
Cf. A032302 (d=2,m=1), A370709 (d=2,m=2), A370716 (d=2,m=3), A370736 (d=2,m=4), A370737 (d=2,m=5).
Cf. A000009 (d=1,m=1), A298994 (d=1,m=2), A303074 (d=1,m=3)

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1+3*x^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1+3*(25*x)^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + 3*(25*x)^k)^(1/5).
a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/5) * 75^n / (5 * Gamma(4/5) * n^(6/5)).

A370761 Expansion of Product_{k>=1} (1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, 5, 14, 70, 196, 640, 2248, 6480, 19072, 56000, 169792, 466560, 1327104, 3642880, 10030080, 27776000, 74541056, 199065600, 531505152, 1401405440, 3672801280, 9674588160, 25018564608, 64701071360, 166363136000, 426159636480, 1084287352832, 2756737761280, 6979072294912
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 01 2024

Keywords

Crossrefs

Cf. A032302, A304961, A370016, A370764, A370765, A370792.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + 2^(k+1)*x^k)*(1 + 2^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (Pi^2/3 + log(2)^2)^(1/4) * 2^(n - 3/4) * exp(sqrt(2*(Pi^2/3 + log(2)^2)*n)) / (3*sqrt(Pi)*n^(3/4)).
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