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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A336979 Expansion of Product_{k>=1} (1 + x^k * (k + x)).

Original entry on oeis.org

1, 1, 3, 6, 11, 21, 37, 69, 108, 192, 312, 522, 827, 1297, 2032, 3240, 4982, 7569, 11508, 17107, 25696, 38340, 57080, 83298, 121373, 175653, 253455, 364307, 523650, 747487, 1063375, 1498471, 2106317, 2955154, 4124071, 5750547, 8000706, 11104596, 15324290, 21093106
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2020

Keywords

Links

Crossrefs

Cf. A022629, A160571, A336975, A336976, A336977, A336978, A336980.

Programs

  • Mathematica
    m = 39; CoefficientList[Series[Product[1 + x^k*(k + x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k*(k+x)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (-1)^(d+1)*(k/d+x)^d/d))))

Formula

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1) * (k/d + x)^d / d).

A267007 Expansion of Product_{k>=1} (1 + (k-1)*x^k).

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 8, 16, 20, 42, 51, 92, 132, 204, 299, 476, 644, 978, 1488, 2024, 3048, 4318, 6248, 8596, 12555, 17378, 24740, 34310, 47940, 65842, 93221, 125238, 173848, 239348, 324724, 445882, 602140, 816424, 1101096, 1495382, 1991892, 2684252, 3598248
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 08 2016

Keywords

Links

Crossrefs

Cf. A022629, A162506, A267004, A267008.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 15 2019
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1+(k-1)*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 0; Do[Do[poly[[j+1]] += (k-1)*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

A300278 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + n*x^n).

Original entry on oeis.org

1, 1, 4, 5, 14, 19, 42, 57, 115, 170, 287, 433, 694, 1061, 1709, 2461, 3740, 5635, 8243, 12256, 18255, 26135, 37826, 54209, 78315, 110488, 159418, 224514, 315414, 442790, 618665, 855640, 1199409, 1642334, 2288904, 3144738, 4303994, 5862294, 8031872, 10869290, 14749050
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 01 2018

Keywords

Comments

Moebius transform of A022629.

Links

Crossrefs

Cf. A000837, A022629, A078374, A085410, A300274, A300275, A300276, A300277.

Programs

  • Mathematica
    nn = 41; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + n x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 41}]

Formula

a(n) = Sum_{d|n} mu(n/d)*A022629(d).

A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).

Original entry on oeis.org

1, 1, 3, 10, 28, 85, 252, 745, 2202, 6530, 19326, 57194, 169341, 501242, 1483816, 4392531, 13002772, 38491212, 113943278, 337298400, 998482338, 2955742400, 8749688247, 25901125616, 76673399424, 226971213462, 671887935923, 1988945626648, 5887744768722, 17429103155892, 51594226501776
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

Comments

Invert transform of A022629.
a(n) is the number of compositions of n where there are A022629(k) sorts of part k. - Joerg Arndt, Jan 24 2024

Links

Crossrefs

Cf. A022629, A299164, A304969, A320652.
Cf. A307057, A307058, A307059, A307060, A307062.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+j*x^j): j in [1..m+2]])) ));
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
  • SageMath
    m=80;
def f(x): return 1/( 2 - product(1+j*x^j for j in range(1,m+3)) )
def A307063_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307063_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A022629(k)*a(n-k).

A032007 "AFK" (ordered, size, unlabeled) transform of 1,2,3,4,...

Original entry on oeis.org

1, 1, 2, 7, 10, 25, 68, 111, 208, 435, 1218, 1773, 3586, 6077, 12156, 31961, 47624, 86825, 151962, 265525, 469610, 1242607, 1750108, 3217663, 5263928, 9205197, 14713474, 26440503, 63610938, 90877893, 159360628, 258871127, 431309688, 687140639, 1134231986
Offset: 0

Views

Author

Christian G. Bower

Keywords

Comments

Sum of products of parts in all compositions of n into distinct parts. - Vladeta Jovovic, Feb 21 2005
Number of compositions of n into distinct parts if there are i kinds of part i. a(3) = 7: 3, 3', 3'', 21, 2'1, 12, 12'.

Links

Crossrefs

Cf. A022629.
Cf. A032005, A032006, A032008.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i<1, 0, b(n, i-1, p)+
      `if`(i>n, 0, i*b(n-i, i-1, p+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 05 2015
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p] + If[i > n, 0, i*b[n - i, i - 1, p + 1]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 11 2019, after Alois P. Heinz *)
  • PARI
    seq(n)={apply(p->subst(serlaplace(p), y, 1), Vec(prod(k=1, n, 1 + k*x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Jun 21 2018

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 05 2015

A266942 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - k*x^k))^k.

Original entry on oeis.org

1, 2, 10, 36, 118, 376, 1156, 3392, 9734, 27230, 74256, 198724, 522292, 1348968, 3432824, 8613856, 21330374, 52190692, 126262774, 302222388, 716247128, 1681575344, 3912919956, 9028823856, 20667406276, 46949343786, 105881451120, 237135574392, 527580701456
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 06 2016

Keywords

Links

Crossrefs

Cf. A006906, A022629, A265758, A266891, A266941.

Programs

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

Formula

From Vaclav Kotesovec, Jan 08 2016: (Start)
a(n) ~ c * n^2 * 3^(n/3), where
c = 1122422673446372185062691708933615715850.583956830118389527... if mod(n,3)=0
c = 1122422673446372185062691708933615715849.484130848291097773... if mod(n,3)=1
c = 1122422673446372185062691708933615715849.782119252925454917... if mod(n,3)=2
(End)

A269144 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - 2*x^k)).

Original entry on oeis.org

1, 3, 10, 29, 77, 195, 475, 1115, 2546, 5706, 12528, 27106, 57893, 122299, 255995, 531816, 1097377, 2252151, 4600835, 9362334, 18990645, 38418370, 77548880, 156251955, 314363615, 631703790, 1268148900, 2543812090, 5099469848, 10217529291, 20464112218
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 20 2016

Keywords

Comments

Convolution of A022629 and A070933.

Links

Crossrefs

Cf. A022629, A070933, A267004, A269153.

Programs

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

Formula

a(n) ~ c * 2^n, where c = Product_{k>=1} (2^k + k)/(2^k - 1) = 19.14883592186082265751161402244824703642181055238186925199088...

A285242 Expansion of Product_{k>=1} (1 + k*x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 4, 13, 9, 0, 0, 36, 36, 0, 16, 52, 63, 27, 64, 64, 108, 108, 64, 233, 277, 135, 27, 676, 676, 108, 204, 772, 1333, 765, 528, 420, 2628, 2628, 528, 1792, 3892, 3735, 1251, 5524, 5380, 4428, 4684, 6657, 12843, 10870, 6703, 3767, 28232
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A022629, A266891, A281790, A285240, A285243, A285244.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + k*x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k, j]*k^j*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x]

A285244 Expansion of Product_{k>=1} (1 + k*x^(k^2)).

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 0, 3, 3, 0, 0, 6, 6, 0, 4, 4, 0, 0, 8, 8, 0, 0, 0, 17, 17, 0, 0, 34, 34, 0, 0, 0, 15, 15, 6, 6, 30, 30, 12, 32, 20, 0, 0, 58, 58, 0, 0, 43, 103, 60, 24, 38, 134, 120, 48, 48, 21, 21, 0, 102, 144, 42, 8, 240, 232, 0, 16, 72, 146, 90, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A022629, A033461, A285242, A285245.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + k*x^(k^2)), {k,1,nmax}], {x,0,nmax}], x]

A292164 Expansion of Product_{k>=1} (1 - k^2*x^k).

Original entry on oeis.org

1, -1, -4, -5, -7, 27, 17, 167, 110, -42, 10, -706, -4001, -3915, 3079, -18640, 9869, 21403, 130565, 107250, -15661, 420664, 599540, -161785, -1232833, -5836888, -5129796, 6516714, -29068180, -14953045, -41490510, 20261320, 30395771, 441235155, 205289550
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2017

Keywords

Links

Crossrefs

Column k=2 of A292166.
Cf. A022629, A022661, A077335, A092484.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
    b[n, i - 1] + If[i > n, 0, i^2*b[n - i, i]]]];
a[n_] := a[n] = If[n == 0, 1,
    -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 04 2024, after Alois P. Heinz *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-n^2*x^n))

Formula

Convolution inverse of A077335.
G.f.: exp(-Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018
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