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 1-9 of 9 results.

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 206, 855, 3585, 15155, 64525, 276278, 1188353, 5130999, 22226049, 96544003, 420368858, 1834203955, 8018057345, 35107961175, 153950675585, 675978772326, 2971700764941, 13078268135683, 57613905606273, 254038914924791, 1121081799217231
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 25 2016

Keywords

Comments

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)

Links

Crossrefs

Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),
       (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[(-1)^j*x^j/(j*(x^j - 1)), {j, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 19 2018 *)
  • PARI
    {a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 26 2019

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

Original entry on oeis.org

5, 3, 5, 2, 7, 0, 1, 3, 3, 3, 4, 8, 6, 6, 4, 2, 6, 8, 7, 7, 7, 2, 4, 1, 5, 8, 1, 4, 1, 6, 5, 3, 2, 7, 8, 7, 9, 8, 5, 1, 4, 8, 3, 2, 7, 1, 2, 8, 6, 9, 4, 7, 0, 9, 7, 3, 1, 9, 6, 9, 0, 7, 5, 6, 0, 6, 4, 1, 0, 2, 1, 5, 1, 2, 6, 7, 5, 3, 1, 5, 5, 2, 2, 3, 2, 3, 4, 2, 7, 6, 4, 4, 7, 8, 8, 5, 4, 2, 2, 8, 2, 2, 8, 1, 7
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 25 2016

Keywords

Examples

			5.352701333486642687772415814165327879851483271286947097319690756...

Links

Crossrefs

Cf. A008485, A109084, A109085, A206228, A270914, A303070, A304625, A327279.

Programs

  • Mathematica
    RealDigits[1/r /. FindRoot[{s == 1/QPochhammer[r*s], QPochhammer[r*s] + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Formula

Equals limit n->infinity A008485(n)^(1/n).

A181315 G.f. A(x) satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^n).

Original entry on oeis.org

1, 1, 2, 6, 19, 64, 227, 832, 3125, 11970, 46579, 183614, 731688, 2942673, 11928707, 48688888, 199932987, 825379993, 3423614756, 14261439594, 59635806865, 250241613688, 1053380320889, 4446989542144, 18823433444211, 79871578901283
Offset: 0

Views

Author

Paul D. Hanna, Oct 16 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 227*x^6 +...
The g.f. A = A(x) satisfies
log(A) = x*A/(1-x^2*A^2) + (x^2/2)*A^2/(1-x^4*A^4) + (x^3/3)*A^3/(1-x^6*A^6) +...

Links

Crossrefs

Cf. A000009, A081362, A109085, A270914, A366018, A366026.

Programs

  • Maple
    nmax:=25: kmax:=nmax: for n from 1 to nmax+1 do A(x):=add(a(k)*x^k, k=0..kmax-1): A(x) := product((1 + x^k*A(x)^k),k=1..kmax+1): a(n-1):=coeff(A(x),x,n-1): od: seq(a(n),n=0..nmax); # Johannes W. Meijer, Jul 04 2011
  • Mathematica
    InverseSeries[x QPochhammer[x, x^2] + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
  • PARI
    {a(n)=polcoeff(1/x*serreverse(x/prod(k=1,n+1,1+x^k+x*O(x^n))),n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x*A+x*O(x^n))^m/(1-(x*A)^(2*m))/m))); polcoeff(A, n)}

Formula

G.f.: A(x) = Sum_{n>=0} A000009(n)*x^n*A(x)^n, where A000009(n) is the number of partitions of n into distinct parts.
G.f.: A(x) = (1/x)*Series_Reversion[x^(1/24)*eta(x)/eta(x^2)] (cf. A081362).
G.f. satisfies A(x) = exp( Sum_{n>=1} (x^n/n)*A(x)^n/(1 - (x*A(x))^(2*n)) ).
a(n) ~ c * d^n / n^(3/2), where d = A270914 = 4.50247674761735448773859393270078440676312875609162163346454... and c = A366018 = 0.482420439587319764659364391266849418507665645926542970519109122... - Vaclav Kotesovec, Aug 21 2018

A206229 a(n) = [x^n] Product_{k=1..n} (1 + x^k)^(n-k+1).

Original entry on oeis.org

1, 1, 2, 8, 31, 124, 515, 2166, 9182, 39195, 168216, 725043, 3136223, 13606891, 59187790, 258034685, 1127137141, 4932071321, 21614913239, 94859273448, 416820578198, 1833626307670, 8074598332650, 35591081565244, 157013886785417, 693237405812328
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2012

Keywords

Examples

			Let [x^n] F(x) denote the coefficient of x^n in F(x); then
a(0) = 1;
a(1) = [x] (1+x) = 1;
a(2) = [x^2] (1+x)^2*(1+x^2) = 2;
a(3) = [x^3] (1+x)^3*(1+x^2)^2*(1+x^3) = 8;
a(4) = [x^4] (1+x)^4*(1+x^2)^3*(1+x^3)^2*(1+x^4) = 31; ...
as illustrated below.
The coefficients in Product_{k=1..n} (1+x^k)^(n-k+1) for n=0..9 begin:
n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
n=1: [1,(1), 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=2: [1, 2,(2), 2, 1, 0, 0, 0, 0, 0, 0, 0 ...];
n=3: [1, 3, 5, (8), 10, 10, 10, 8, 5, 3, 1, 0 ...];
n=4: [1, 4, 9, 18, (31), 46, 64, 82, 96, 106, 110, 106 ...];
n=5: [1, 5, 14, 33, 68, (124), 210, 332, 492, 693, 931, ...];
n=6: [1, 6, 20, 54, 127, 266, (515), 934, 1597, 2602, ...];
n=7: [1, 7, 27, 82, 215, 502, 1078, (2166), 4109, 7428, ...];
n=8: [1, 8, 35, 118, 340, 870, 2038, 4454, (9182), 18020, ...];
n=9: [1, 9, 44, 163, 511, 1417, 3582, 8420, 18634,(39195), ...]; ...
where the coefficients in parenthesis start this sequence.

Links

Crossrefs

Cf. A206228.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + x^k)^(n-k+1), {k, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Aug 21 2018 *)
  • PARI
    {a(n)=polcoeff(prod(k=1,n,(1+x^k+x*O(x^n))^(n-k+1)),n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007844067631287560916216334645404240888403... and c = 0.1630284922981520921416997097273846855003438911417350833863798... - Vaclav Kotesovec, Aug 21 2018

A327214 Self-convolution of A270913.

Original entry on oeis.org

1, 2, 7, 32, 137, 592, 2597, 11442, 50567, 224112, 995392, 4428372, 19727877, 87983202, 392755207, 1754625632, 7844003907, 35086658052, 157023432677, 703037135122, 3148915010832, 14108913792342, 63235380631747, 283495965998772, 1271282293531077, 5702105357347602
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 26 2019

Keywords

Links

Crossrefs

Cf. A022567, A270913, A304443, A309986, A327215.

Programs

  • Mathematica
    A270913[n_]:=SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A270913[k]*A270913[n-k], {k, 0, n}], {n, 0, 25}]

Formula

a(n) ~ c^2 * Pi * d^n, where d = A270914 = 4.5024767476173544877385939327... and c = A327280 = 0.260542233142438469433860832160... (see A270913).

A327280 Decimal expansion of a constant related to A270913 and A327214.

Original entry on oeis.org

2, 6, 0, 5, 4, 2, 2, 3, 3, 1, 4, 2, 4, 3, 8, 4, 6, 9, 4, 3, 3, 8, 6, 0, 8, 3, 2, 1, 6, 0, 4, 2, 2, 4, 8, 7, 6, 7, 6, 9, 0, 2, 0, 6, 0, 3, 9, 0, 8, 1, 6, 3, 0, 9, 2, 6, 3, 0, 2, 5, 9, 0, 1, 0, 2, 4, 1, 0, 3, 7, 5, 7, 2, 8, 9, 6, 2, 5, 4, 2, 2, 7, 8, 3, 5, 6, 5, 0, 8, 6, 0, 4, 0, 4, 4, 3, 1, 1, 4, 1, 0, 1, 0, 6, 2, 6
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2019

Keywords

Examples

			0.26054223314243846943386083216042248767690206039081630926302590102410375728962...

Crossrefs

Cf. A270913, A270914, A327214, A327279, A366018.

Programs

  • Mathematica
    RealDigits[1/Sqrt[s*Pi*Derivative[0, 2][QPochhammer][-1, r*s]]/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 02 2023 *)

Formula

Equals limit_{n->infinity} A270913(n) * sqrt(n) / A270914^n.

A303071 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + x^k)^n.

Original entry on oeis.org

1, 2, 6, 23, 90, 362, 1491, 6225, 26242, 111479, 476466, 2046464, 8825559, 38191467, 165751529, 721177328, 3144703234, 13739010855, 60127642329, 263545670385, 1156732481150, 5083320593976, 22364017244278, 98491038664903, 434160710647831, 1915482295831037, 8457663096970431
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2018

Keywords

Links

Crossrefs

Cf. A036469, A270913, A286335, A303070.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - x) Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Exp[n Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

Formula

a(n) = [x^n] (1/(1 - x))*exp(n*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).
a(n) = Sum_{j=0..n} A286335(j,n).
a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = 0.44252758868364961050787771300805... - Vaclav Kotesovec, May 19 2018

A304626 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(n*k)))^n.

Original entry on oeis.org

1, 0, 1, 10, 47, 201, 849, 3578, 15147, 64516, 276268, 1188342, 5130987, 22226036, 96543989, 420368843, 1834203939, 8018057328, 35107961157, 153950675566, 675978772306, 2971700764920, 13078268135661, 57613905606250, 254038914924767, 1121081799217206, 4951199308679965
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Comments

Number of partitions of n into 2 or more distinct parts, with n types of each part. - Ilya Gutkovskiy, May 16 2018

Links

Crossrefs

Cf. A058576, A073252, A111133, A255526, A270913, A304625.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(n k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(QPochhammer[-1, x, 1 + n]/QPochhammer[-1, x^n, 1 + n])^n, {x, 0, n}], {n, 0, 26}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.502476747617354487738... and c = 0.2605422331424384694... - Vaclav Kotesovec, May 16 2018

A366018 Decimal expansion of a constant related to the asymptotics of A181315.

Original entry on oeis.org

4, 8, 2, 4, 2, 0, 4, 3, 9, 5, 8, 7, 3, 1, 9, 7, 6, 4, 6, 5, 9, 3, 6, 4, 3, 9, 1, 2, 6, 6, 8, 4, 9, 4, 1, 8, 5, 0, 7, 6, 6, 5, 6, 4, 5, 9, 2, 6, 5, 4, 2, 9, 7, 0, 5, 1, 9, 1, 0, 9, 1, 2, 2, 0, 4, 3, 4, 1, 3, 7, 3, 9, 5, 8, 5, 8, 6, 9, 4, 1, 9, 5, 1, 6, 2, 8, 2, 6, 4, 5, 6, 5, 6, 1, 8, 0, 6, 5, 8, 4, 9, 1, 8, 9, 1, 8
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 26 2023

Keywords

Examples

			0.482420439587319764659364391266849418507665645926542970519109122...

Crossrefs

Cf. A181315, A270914.

Programs

  • Mathematica
    RealDigits[Sqrt[s/(Pi*Derivative[0, 2][QPochhammer][-1, r*s])]/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]]

Formula

Equals limit_{n->infinity} A181315(n) * n^(3/2) / A270914^n.
Showing 1-9 of 9 results.

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