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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A307259 Expansion of (1/(1 - x)) * Product_{k>=1} (1 + k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 2, 5, 15, 44, 126, 357, 1003, 2783, 7618, 20627, 55421, 148021, 393140, 1038123, 2724992, 7112022, 18465708, 47726767, 122861732, 315123476, 805428727, 2051556778, 5207982062, 13177117709, 33235023381, 83574705456, 209576713721, 524181331710, 1307849984089, 3255539133109
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

Binomial transform of A022629.

Crossrefs

Cf. A022629, A266232, A294502, A307260, A318127.

Programs

  • Maple
    a:=series((1/(1-x))*mul(1+k*x^k/(1-x)^k,k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - x) Product[(1 + k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*A022629(k).

A307260 Expansion of (1/(1 + x)) * Product_{k>=1} (1 + k*x^k/(1 + x)^k).

Original entry on oeis.org

1, 0, 1, 1, -4, 14, -35, 77, -161, 356, -873, 2267, -5787, 13850, -30361, 59934, -103754, 147968, -139049, -58998, 730972, -2430881, 6333238, -15548722, 39845197, -110775861, 325257904, -960503811, 2756222486, -7568564555, 19815541729, -49548068461, 118752506024
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

Inverse binomial transform of A022629.

Crossrefs

Cf. A022629, A293467, A294503, A307258, A307259.

Programs

  • Maple
    a:=series((1/(1+x))*mul(1+k*x^k/(1+x)^k,k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 + x) Product[(1 + k x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A022629(k).

A318484 Expansion of Product_{k>=1} (1 + k*x^k)^sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 1, 6, 18, 52, 142, 404, 1018, 2624, 6645, 16124, 38857, 92245, 214841, 494098, 1125062, 2522188, 5604930, 12327860, 26838595, 57913194, 123951482, 263019720, 553989989, 1158449522, 2405179547, 4961047246, 10168544537, 20714279168, 41952595411, 84494479578
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 27 2018

Keywords

Links

Crossrefs

Cf. A000203, A022629, A266891, A318416, A107742, A192065, A318483.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1+k*x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k], j]*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[s, x]

A333653 Expansion of Product_{i>=1, j>=1} (1 + i*x^(i*j)).

Original entry on oeis.org

1, 1, 3, 7, 13, 27, 54, 98, 174, 335, 572, 1004, 1733, 2933, 4916, 8307, 13470, 22042, 35851, 57256, 91462, 145231, 227667, 355522, 554058, 853986, 1313121, 2010318, 3057827, 4627213, 6989808, 10481205, 15679549, 23365207, 34658909, 51241077, 75541695, 110852295, 162238415
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2020

Keywords

Links

Crossrefs

Cf. A022629, A107742, A285244, A318416, A332199.

Programs

  • Mathematica
    m = 38; CoefficientList[Series[Product[1 + i*x^(i*j), {i, 1, m}, {j, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 23 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(prod(i=1, N, prod(j=1, N\i, 1+i*x^(i*j))))
  • PARI
    N=40; x='x+O('x^N); Vec(prod(k=1, N, prod(d=1, k, 1+(k%d==0)*d*x^k)))

Formula

G.f.: Product_{k>0} f(q^k) where f(q) = Product_{i>=1} (1 + i*q^i).
G.f.: Product_{k>0} Product_{d|k} (1 + d*x^k).

A344370 Dirichlet g.f.: Product_{k>=2} (1 + k^(1-s)).

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 7, 16, 9, 20, 11, 36, 13, 28, 30, 32, 17, 54, 19, 60, 42, 44, 23, 120, 25, 52, 54, 84, 29, 150, 31, 96, 66, 68, 70, 180, 37, 76, 78, 200, 41, 210, 43, 132, 135, 92, 47, 336, 49, 150, 102, 156, 53, 270, 110, 280, 114, 116, 59, 540
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2021

Keywords

Crossrefs

Cf. A022629, A045778, A050368, A050369, A224892, A344368, A344369.

Programs

  • Mathematica
    T[m_, 1] := 1; T[1, n_] := 0; T[1, 1] := 1; T[0, n_] := 0; T[m_, n_] := T[m, n] = Total[T[# - 1, n/#] & /@ Select[Divisors[n], # <= m &]]; A045778[n_] := T[n, n]; Table[n A045778[n], {n, 60}]

Formula

a(n) = n * A045778(n).

A282207 Expansion of Product_{k>=0} (1 + (2*k + 1)*x^(2*k+1)).

Original entry on oeis.org

1, 1, 0, 3, 3, 5, 5, 7, 22, 24, 30, 32, 73, 75, 91, 198, 277, 309, 339, 560, 689, 1078, 1126, 1567, 2703, 3396, 3676, 5086, 7046, 8241, 10896, 13072, 19891, 22975, 27922, 41597, 56117, 62459, 77183, 100793, 131846, 161665, 191446, 255225, 311247, 408418, 467460, 599970, 843441
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 09 2017

Keywords

Comments

Sum of products of terms in all partitions of n into distinct odd parts.

Examples

			a(10) = 30 because we have [9, 1], [7, 3], 9*1 = 9, 7*3 = 21 and 9 + 21 = 30.

Links

Crossrefs

Cf. A000700, A022629, A067553.

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[Product[1 + (2 k + 1) x^(2 k + 1), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=0} (1 + (2*k + 1)*x^(2*k+1)).

A282866 Expansion of Product_{k>=1} (1 + k^2*x^(k^2)).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 0, 9, 9, 0, 0, 36, 36, 0, 16, 16, 0, 0, 64, 64, 0, 0, 0, 169, 169, 0, 0, 676, 676, 0, 0, 0, 225, 225, 36, 36, 900, 900, 144, 544, 400, 0, 0, 1924, 1924, 0, 0, 1345, 4945, 3600, 576, 772, 14596, 14400, 2304, 2304, 441, 441, 0, 6084, 7848, 1764, 64, 25184, 25120, 0, 256, 3392, 11236, 8100, 0, 576
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 23 2017

Keywords

Comments

Sum of products of terms in all partitions of n into distinct squares (A000290).

Examples

			a(41) = 544 because we have [36, 4, 1], [25, 16], 36*4*1 = 144, 25*16 = 400 and 144 + 400 = 544.

Links

Crossrefs

Cf. A000290, A022629, A033461, A282209.

Programs

  • Mathematica
    nmax = 73; CoefficientList[Series[Product[1 + k^2 x^k^2, {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    Vec(prod(k=1, 73, (1 + k^2*x^(k^2))) + O(x^73)) \\ Indranil Ghosh, Mar 15 2017

Formula

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

A294464 Expansion of e.g.f. Product_{k>0} (1+k*x^k)^(1/k).

Original entry on oeis.org

1, 1, 2, 12, 36, 300, 2520, 20160, 75600, 2192400, 30996000, 276091200, 2165486400, 19070251200, 968042275200, 41954552640000, 190974944160000, 230641066656000, 95607669148992000, -2052972258809472000, 22839078791168640000, 5074390517301705600000
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Crossrefs

Cf. A022629, A076717, A168243, A294462.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+k*x^k)^(1/k))))

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018
a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} A076717(k)*a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jan 21 2025

A300786 L.g.f.: log(Product_{k>=1} (1 + k*x^k)) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 3, 10, 7, 26, 24, 50, -33, 163, 38, 122, -188, 170, 108, 1580, -1793, 290, -273, 362, -1678, 9404, 3248, 530, -49092, 16251, 14862, 66340, 14000, 842, -135556, 962, -429057, 547172, 258386, 509500, -1392821, 1370, 1043160, 4813052, -8088838, 1682, -9267612, 1850, 8218844, 53396438
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 12 2018

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 7*x^4/4 + 26*x^5/5 + 24*x^6/6 + 50*x^7/7 - 33*x^8/8 + 163*x^9/9 + 38*x^10/10 + ...
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 15*x^5 + 25*x^6 + 43*x^7 + 64*x^8 + 120*x^9 + 186*x^10 + ... + A022629(n)*x^n + ...

Crossrefs

Cf. A022629, A076717, A078308.

Programs

  • Mathematica
    nmax = 45; Rest[CoefficientList[Series[Log[Product[(1 + k x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[k^2 x^k/(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[(-d)^(n/d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 45}]

Formula

L.g.f.: Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j = Sum_{n>=1} a(n)*x^n/n.
G.f.: Sum_{k>=1} k^2*x^k/(1 + k*x^k).
a(n) = Sum_{d|n} (-d)^(n/d+1).

A302289 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + k*x^k*A(x)).

Original entry on oeis.org

1, 1, 3, 10, 30, 98, 323, 1083, 3684, 12710, 44272, 155608, 551259, 1965952, 7052990, 25436711, 92168542, 335376653, 1224991077, 4489818110, 16507728007, 60868469848, 225030777305, 833961333273, 3097594423724, 11529400593846, 42996077073284, 160632616725238, 601132116489719, 2253153800577748
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 04 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 3*x^2 + 10*x^3 + 30*x^4 + 98*x^5 + 323*x^6 + 1083*x^7 + 3684*x^8 + 12710*x^9 + 44272*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + 2*x^2*A(x)) * (1 + 3*x^3*A(x)) * (1 + 4*x^4*A(x)) * ...

Crossrefs

Cf. A022629, A145267, A301456, A301577, A301578, A302287, A302288.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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