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.

Showing 1-7 of 7 results.

A007312 Reversion of g.f. (with constant term omitted) for partition numbers.

Original entry on oeis.org

1, -2, 5, -15, 52, -200, 825, -3565, 15900, -72532, 336539, -1582593, 7524705, -36111810, 174695712, -851020367, 4171156249, -20555470155, 101787990805, -506227992092, 2527493643612, -12663916942984, 63656297034920, -320914409885850, 1622205233276889
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000041, A050393, A066398, A334315.

Programs

  • Maple
    # Using function CompInv from A357588.
CompInv(25, n -> combinat:-numbpart(n)); # Peter Luschny, Oct 05 2022
  • Mathematica
    nmax = 30; Rest[CoefficientList[InverseSeries[Series[Sum[PartitionsP[n]*x^n, {n, 1, nmax}], {x, 0, nmax}]], x]] (* Vaclav Kotesovec, Nov 11 2017 *)
Rest[CoefficientList[InverseSeries[Series[-1 + 1/QPochhammer[x],{x,0,30}],x],x]] (* Vaclav Kotesovec, Jan 18 2024 *)
(* Calculation of constant d: *) Chop[1/r /. FindRoot[{(1 + r)*QPochhammer[s, s] == 1, Log[1 - s] + QPolyGamma[0, 1, s] - (1 + r)*s*Log[s] * Derivative[0, 1][QPochhammer][s, s] == 0}, {r, -1/5}, {s, -1/2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 18 2024 *)

Formula

From Vaclav Kotesovec, Nov 11 2017: (Start)
a(n) ~ -(-1)^n * c * d^n / n^(3/2), where
d = 5.379264118840884783404842050140885100801253519243086... and
c = 0.10697042824132534557642152089737206588353695053... (End)
G.f. A(x) satisfies: A(x) = 1 - (1/(1 + x)) * Product_{k>=2} 1/(1 - A(x)^k). - Ilya Gutkovskiy, Apr 23 2020

Extensions

Signs corrected Dec 24 2001

A050394 Exponential reversion of partitions into distinct parts A000009.

Original entry on oeis.org

1, -1, 1, 3, -38, 234, -586, -9493, 194906, -1981880, 4724642, 301409600, -7840250579, 102256372812, -9254171165, -39936103724491, 1199830608004118, -17733202003472076, -96125957089130420
Offset: 1

Views

Author

Christian G. Bower, Nov 15 1999

Keywords

Links

Crossrefs

Cf. A050393.

Programs

  • Mathematica
    length = 20; Range[length]! InverseSeries[Sum[PartitionsQ[n] x^n/n!, {n, 1, length}] + O[x]^(length+1)][[3]] (* Vladimir Reshetnikov, Nov 07 2015 *)

A291489 Expansion of the series reversion of -1 + Product_{k>=1} (1 + x^k)^k.

Original entry on oeis.org

1, -2, 3, 2, -41, 196, -541, 229, 7235, -48228, 175956, -254933, -1575661, 14909191, -67194669, 153944915, 292516673, -4968647665, 27275432639, -82747735226, 3883854725, 1660136515050, -11302429310683, 42362000190568, -53376259124482, -520085199830413, 4671353423344131
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A026007.

Links

Crossrefs

Cf. A007312, A026007, A050393, A176025.

Programs

  • Mathematica
    nmax = 27; Rest[CoefficientList[InverseSeries[Series[-1 + Product[(1 + x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 27; Rest[CoefficientList[InverseSeries[Series[-1 + Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x], x]]

Formula

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

A291488 Expansion of the series reversion of -1 + Product_{k>=1} 1/(1 - x^k)^k.

Original entry on oeis.org

1, -3, 12, -58, 318, -1896, 11966, -78595, 531486, -3674324, 25845131, -184348434, 1330147092, -9690872427, 71189146313, -526703176813, 3921274277132, -29354616797397, 220824254874928, -1668453804382315, 12655766723174710, -96340024533522759, 735747052686408916, -5635489764030599334
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A000219.

Links

Crossrefs

Cf. A000219, A007312, A050393, A176025.

Programs

  • Mathematica
    nmax = 24; Rest[CoefficientList[InverseSeries[Series[-1 + Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 24; Rest[CoefficientList[InverseSeries[Series[-1 + Exp[Sum[DivisorSigma[2, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x], x]]

Formula

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

A291645 Expansion of the series reversion of -1 + Product_{k>=1} (1 + x^(k^2)).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, 4, 9, 4, -23, -78, -78, 132, 694, 1088, -443, -6169, -13452, -4646, 52247, 155891, 143796, -391672, -1715015, -2481013, 2107735, 17836000, 35704800, 3037215, -172386166, -465009936, -338007604, 1487272659, 5624864403, 7125599375, -10208041482
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A033461.

Links

Crossrefs

Cf. A007312, A033461, A050393, A291489.

Programs

  • Mathematica
    nmax = 36; Rest[CoefficientList[InverseSeries[Series[-1 + Product[1 + x^k^2, {k, 1, nmax}], {x, 0, nmax}], x], x]]

Formula

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

A291646 Expansion of the series reversion of -1 + Product_{k>=1} (1 + x^(2*k-1)).

Original entry on oeis.org

1, 0, -1, -1, 2, 6, -1, -29, -32, 108, 311, -185, -1991, -1590, 9468, 22163, -26645, -170511, -70359, 955734, 1755790, -3561052, -16020532, 309754, 102695477, 141637053, -463468990, -1567907433, 806541136, 11367276801, 10768399120, -59447130815, -155142592628, 172852194214, 1273466836673
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A000700.

Links

Crossrefs

Cf. A000700, A007312, A050393, A291489.

Programs

  • Mathematica
    nmax = 35; Rest[CoefficientList[InverseSeries[Series[-1 + Product[1 + x^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 35; Rest[CoefficientList[InverseSeries[Series[-1 + QPochhammer[x^2]^2/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x], x]]

Formula

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

A291695 Expansion of the series reversion of Sum_{i>=1} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).

Original entry on oeis.org

1, -3, 12, -57, 304, -1757, 10746, -68450, 449274, -3016645, 20618317, -142946735, 1002722249, -7103064540, 50738237140, -365049115546, 2642981328372, -19241453032254, 140770867457795, -1034409857616986, 7631075823632553, -56497364856268721, 419641611512419630, -3126180409889288924
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

Comments

Reversion of g.f. for A006128.

Links

Crossrefs

Cf. A006128, A007312, A050393, A176025.

Programs

  • Mathematica
    nmax = 24; Rest[CoefficientList[InverseSeries[Series[Sum[x^i/(1 - x^i), {i, 1, nmax}] / Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 24; Rest[CoefficientList[InverseSeries[Series[(Log[1-x] + QPolyGamma[0, 1, x]) / (Log[x]*QPochhammer[x]), {x, 0, nmax}], x], x]] (* Vaclav Kotesovec, Apr 21 2020 *)

Formula

G.f. A(x) satisfies: Sum_{i>=1} A(x)^i/(1 - A(x)^i) / Product_{j>=1} (1 - A(x)^j) = x.
G.f. A(x) satisfies: Sum_{i>=1} i*A(x)^i / Product_{j=1..i} (1 - A(x)^j) = x.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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