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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A300187 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k)^(n/k).

Original entry on oeis.org

1, 1, 4, 39, 488, 7615, 147024, 3371137, 89079808, 2665537713, 89142430400, 3295096700071, 133399600068096, 5870116973678191, 278971698167158528, 14239859507270510625, 776985219329347518464, 45130494178637796970273, 2780224621391401396134912, 181059775626543107582734183
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 + x^k)^(n/k) begins:
n = 0: (1), 0,   0,    0,     0,      0,       0,  ...
n = 1:  1, (1),  1,    5,    11,     59,     439,  ...
n = 2:  1,  2,  (4),  16,    68,    328,    2416,  ...
n = 3:  1,  3,   9,  (39),  207,   1197,    8811,  ...
n = 4:  1,  4,  16,   80,  (488),  3296,   25984,  ...
n = 5:  1,  5,  25,  145,   995,  (7615),  65575,  ...
n = 6:  1,  6,  36,  240,  1836,  15624, (147024), ...

Links

Crossrefs

Cf. A048272, A168243, A270922, A299033, A299034, A300188.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

Formula

a(n) = n! * [x^n] exp(n*Sum_{k>=1} A048272(k)*x^k/k).
a(n) ~ c * d^n * n^n, where d = 1.294982800733109500251... and c = 0.6755467963500480915... - Vaclav Kotesovec, Sep 07 2018

A304494 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^H(k), where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 1, 3, 20, 103, 899, 8143, 84678, 975049, 13082993, 186340631, 2878977408, 48899305783, 876721463435, 16971889682707, 349059348881834, 7565120836998801, 173313418321443809, 4197655086606145387, 106097089652021765356, 2816940203630838490791, 78147038018470085005235
Offset: 0

Views

Author

Ilya Gutkovskiy, May 13 2018

Keywords

Links

Crossrefs

Cf. A001008, A002805, A168243, A303970, A304496.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[(1 + x^k)^HarmonicNumber[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d HarmonicNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(A001008(k)/A002805(k)).

A318967 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(1/(i*j*k)).

Original entry on oeis.org

1, 1, 3, 15, 69, 477, 4167, 34731, 333225, 4058073, 48535659, 638782119, 9690930477, 146665611765, 2428164153711, 44904494549763, 820664075440593, 16238018609968689, 350155700132388435, 7568774583230565567, 175171222712837235861, 4318996957424273510541, 107317465474650443023383
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

Links

Crossrefs

Cf. A000005, A007425, A168243, A280473, A318414, A318696, A318768, A318966.

Programs

  • Maple
    a:=series(mul(mul(mul((1+x^(i*j*k))^(1/(i*j*k)),k=1..55),j=1..55),i=1..55),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[(-1)^(k/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(tau_3(k)/k), where tau_3 = A007425.
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1) * Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).

A338813 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} (1+x^j)^(u/j).

Original entry on oeis.org

1, 0, 1, 4, 0, 1, -6, 16, 0, 1, 48, -30, 40, 0, 1, 0, 448, -90, 80, 0, 1, 1440, -840, 2128, -210, 140, 0, 1, -10080, 23532, -6720, 7168, -420, 224, 0, 1, 120960, -127008, 177868, -30240, 19488, -756, 336, 0, 1, 0, 2191104, -1018080, 892540, -100800, 45696, -1260, 480, 0, 1
Offset: 1

Views

Author

Seiichi Manyama, Nov 10 2020

Keywords

Comments

Also the Bell transform of A338814.

Examples

			exp(Sum_{n>0} u*A048272(n)*x^n/n) = 1 + u*x + u^2*x^2/2! + (4*u+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       0,     1;
       4,     0,     1;
      -6,    16,     0,    1;
      48,   -30,    40,    0,    1;
       0,   448,   -90,   80,    0,   1;
    1440,  -840,  2128, -210,  140,   0, 1;
  -10080, 23532, -6720, 7168, -420, 224, 0, 1;
  ...

Links

Crossrefs

Column k=1 gives A338814.
Row sums give A168243.
Cf. A048272, A075525, A338805.

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)
  • PARI
    {T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1+x^j+x*O(x^n))^(u/j)), n), k)}
  • PARI
    a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

Formula

E.g.f.: exp(Sum_{n>0} u*A048272(n)*x^n/n).
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} A048272(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} A048272(i_j)/i_j.

A345750 E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k).

Original entry on oeis.org

1, 1, 2, 9, 49, 310, 2521, 25557, 290550, 3555041, 48104901, 741103946, 12825399313, 240202011881, 4747281446090, 98808864563065, 2194031697420057, 52582450760730398, 1357237338948268649
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2021

Keywords

Comments

Stirling transform of A168243.

Links

Crossrefs

Cf. A048272, A048993, A168243, A305550, A305987, A345749, A345751.

Programs

  • Mathematica
    max = 18; Range[0, max]! * CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k)^(1/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+(exp(x)-1)^k)^(1/k))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, -sumdiv(k, d, (-1)^d)*(exp(x)-1)^k/k))))

Formula

E.g.f.: exp( Sum_{k>=1} A048272(k) * (exp(x) - 1)^k / k ).
a(n) = Sum_{k=0..n} Stirling2(n,k) * A168243(k).

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

A295833 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^k/k).

Original entry on oeis.org

1, -1, 3, -11, 47, -279, 2089, -16057, 137409, -1417553, 15656651, -187422531, 2501688463, -34832785831, 529520417217, -8723102543009, 146573712239489, -2670058109819937, 52017332039568019, -1041334898093864443, 22335551258991482991, -502509800119879530551, 11641825391540821682393
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Examples

			E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...)/((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...) = 1 - x + 3*x^2/2! - 11*x^3/3! + 47*x^4/4! - 279*x^5/5! + 2089*x^6/6! - 16057*x^7/7! + ...

Crossrefs

Cf. A028342, A168243, A206303, A284474, A294356, A295792, A295834.

Programs

  • Maple
    a:=series(mul((1+x^k)^((-1)^k/k),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k)^((-1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A295834 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^(k+1)/k).

Original entry on oeis.org

1, 1, -1, -1, 11, 19, -311, -1919, 20201, 154169, -1363249, -14236289, 140759299, 1213688059, -33239720359, -257577468511, 11707385639249, 119005356808561, -3416942071608929, -43117983466829441, 893917358612502011, 13133282766425234531, -411010168576899605911, -7970128344774479644991
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Examples

			E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...)/((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...) = 1 + x - x^2/2! - x^3/3! + 11*x^4/4! + 19*x^5/5! - 311*x^6/6! - 1919*x^7/7! + ...

Crossrefs

Cf. A028342, A168243, A206303, A284467, A294356, A295792, A295833.

Programs

  • Maple
    a:=series(mul((1+x^k)^((-1)^(k+1)/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((-1)^(k+1)/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A307680 Expansion of e.g.f. Product_{k>=1} (1 + x^k/(1 - x)^k)^(1/k).

Original entry on oeis.org

1, 1, 3, 17, 131, 1239, 14029, 187627, 2906553, 50982929, 993806531, 21270277401, 496425262123, 12577053063847, 344382608381421, 10139294386051139, 319175215666010609, 10684742192933940897, 378662321114852778883, 14158327369578651838369, 557151639159864934384851
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 21 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 131*x^4/4! + 1239*x^5/5! + 14029*x^6/6! + 187627*x^7/7! + 2906553*x^8/8! + ...

Crossrefs

Cf. A048272, A129519, A168243, A307679, A320564.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(Sum_{k>=1} A048272(k)*x^k/(k*(1 - x)^k)).
a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A168243(k)*n!/k!.

A352003 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k))^(1/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, -3, 44, -35, 1014, -1127, 46808, 153081, 3240170, -30922859, 443621892, 331421077, 121899383774, 691635821745, 19657393214384, 424491327098353, 2132527815161298, -2864544697983059, 3885322666246386140, 22621061924336157261, 882556261002776755142
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2022

Keywords

Crossrefs

Cf. A168243, A305614, A318913, A352005.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+isprime(k)*x^k)^(1/k))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, isprime(d)*(-1)^(k/d+1))*x^k/k))))

Formula

E.g.f.: exp( Sum_{k>=1} A305614(k)*x^k/k ) where A305614(k) = Sum_{p|k, p prime} (-1)^(k/p+1).
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