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.

Previous Showing 11-16 of 16 results.

A317017 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k+1)*binomial(k+2,3)/4).

Original entry on oeis.org

1, 1, 8, 33, 126, 441, 1571, 5338, 17900, 58359, 187134, 588966, 1826537, 5580784, 16831549, 50135506, 147650112, 430187724, 1240908651, 3545808444, 10042128414, 28201458999, 78567720054, 217225969695, 596254164090, 1625343030654, 4401332943214, 11843216471115, 31674767502610
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2018

Keywords

Comments

Euler transform of A001296.

Links

Crossrefs

Cf. A000391, A001296, A278768, A279216, A305653, A317019, A317020, A317021.

Programs

  • Maple
    a:=series(mul(1/(1-x^k)^((3*k+1)*binomial(k+2,3)/4),k=1..100),x=0,29): seq(coeff(a,x,n),n=0..28); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((3 k + 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + 2 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d + 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A001296(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^k)^5)).
a(n) ~ Pi^(1/288)/(2 * 3^(577/864) * 7^(145/1728) * n^(1009/1728)) * exp(1/144 - (1/12-Zeta'(-1))/12 - (11 * Zeta(3))/(80 * Pi^2) + (1383 * Zeta(5))/(640 * Pi^4) + (11025 * Zeta(3) * Zeta(5)^2)/(2 * Pi^12) - (694575 * Zeta(5)^3)/(2 * Pi^14) + (13127467500 * Zeta(5)^5)/Pi^24 + (5 * Zeta'(-3))/12 + ((-21 * 3^(1/3) * 7^(1/6) * Pi)/6400 - (35 * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/(2 * Pi^7) + (15435 * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(16 * Pi^9) - (175573125 * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/(4 * Pi^19)) * n^(1/6) + (((7/3)^(1/3) * Zeta(3))/(4 * Pi^2) - (21 * 3^(2/3) * 7^(1/3) * Zeta(5))/(8 * Pi^4) + (147000 * 3^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7) * Pi)/40 - (1575 * sqrt(7) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((15 * 3^(1/3) * 7^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * 3^(2/3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A317019 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*binomial(k+2,3)).

Original entry on oeis.org

1, 1, 9, 39, 155, 570, 2131, 7599, 26667, 90996, 305144, 1004173, 3254123, 10385884, 32704819, 101678860, 312435675, 949498206, 2855953018, 8507079361, 25108844890, 73468004480, 213201630328, 613871526178, 1754365814430, 4978113020152, 14029639217532, 39281646364737
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2018

Keywords

Comments

Euler transform of A002417.

Links

Crossrefs

Cf. A000391, A002417, A278767, A279217, A305653, A317017, A317020, A317021.

Programs

  • Maple
    a:=series(mul(1/(1-x^k)^(k*binomial(k+2,3)),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 3 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^3 (d + 1) (d + 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002417(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 3*x^k)/(k*(1 - x^k)^5)).
a(n) ~ 1/(2^(601/720) * 3^(359/480) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)) * exp(-Zeta(3)/(12 * Pi^2) + (491 * Zeta(5))/(400 * Pi^4) - (2250423 * Zeta(5)^3)/(10 * Pi^14) + (103355177121 * Zeta(5)^5)/(10 * Pi^24) + Zeta'(-3)/2 + ((-7 * 7^(1/6) * Pi)/(1200 * 2^(1/3) * sqrt(3)) + (27783 * sqrt(3) * 7^(1/6) * Zeta(5)^2)/(40 * 2^(1/3) * Pi^9) - (614365479 * sqrt(3) * 7^(1/6) * Zeta(5)^4)/(16 * 2^(1/3) * Pi^19)) * n^(1/6) + ((-63 * 7^(1/3) * Zeta(5))/(10 * 2^(2/3) * Pi^4) + (214326 * 14^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/3) * Pi)/30 - (1701 * sqrt(21) * Zeta(5)^2)/(2 * Pi^9)) * sqrt(n) + ((27 * 7^(2/3) * Zeta(5))/(2 * 2^(1/3) * Pi^4)) * n^(2/3) + ((2 * 2^(1/3) * sqrt(3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5*k-1)*binomial(k+2,3)/4).

Original entry on oeis.org

1, 1, 10, 45, 185, 710, 2766, 10270, 37444, 132765, 462327, 1579563, 5311361, 17584084, 57414594, 185032557, 589183035, 1854974066, 5778722817, 17823440534, 54458411332, 164917654587, 495219323844, 1475145786950, 4360576440676, 12796007418881, 37287660835368, 107930276062786
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2018

Keywords

Comments

Euler transform of A002418.

Links

Crossrefs

Cf. A000391, A002418, A278769, A279218, A305653, A317017, A317019, A317021.

Programs

  • Maple
    a:=series(mul(1/(1-x^k)^((5*k-1)*binomial(k+2,3)/4),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^((5 k - 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 4 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (5 d - 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002418(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 4*x^k)/(k*(1 - x^k)^5)).
a(n) ~ (5/7)^(703/8640)/(2 * 3^(2143/2880) * n^(5023/8640) * Pi^(17/1440)) * exp(-1/144 + (1/12-Zeta'(-1))/12 - (21 * Zeta(3))/(400 * Pi^2) + (62921 * Zeta(5))/(80000 * Pi^4) - (194481 * Zeta(3) * Zeta(5)^2)/(50 * Pi^12) - (200120949 * Zeta(5)^3)/(1250 * Pi^14) + (28594081676916 * Zeta(5)^5)/(3125 * Pi^24) + (7 * Zeta'(-3))/12 + ((-343 * (7/5)^(1/6) * Pi)/(96000 * sqrt(3)) + (147 * (7/5)^(1/6) * sqrt(3) * Zeta(3) * Zeta(5))/(10 * Pi^7) + (1058841 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^2)/(2000 * Pi^9) - (18211006359 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^4)/(500 * Pi^19)) * n^(1/6) + (-((7/5)^(1/3) * Zeta(3))/(4 * Pi^2) - (1029 * (7/5)^(1/3) * Zeta(5))/(200 * Pi^4) + (10890936 * (7/5)^(1/3) * Zeta(5)^3)/(25 * Pi^14)) * n^(1/3) + ((7 * sqrt(7/15) * Pi)/120 - (9261 * sqrt(21/5) * Zeta(5)^2)/(5 * Pi^9)) * sqrt(n) + ((63 * (7/5)^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * sqrt(3) * Pi)/(5^(5/6) * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A317021 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k-1)*binomial(k+2,3)/2).

Original entry on oeis.org

1, 1, 11, 51, 216, 861, 3477, 13367, 50377, 184667, 664484, 2345230, 8142476, 27825576, 93750686, 311682789, 1023547782, 3322634928, 10669887669, 33916213669, 106776876109, 333111724130, 1030264525744, 3160359629535, 9618807643826, 29057370625281, 87153154537437
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2018

Keywords

Comments

Euler transform of A002419.

Links

Crossrefs

Cf. A000391, A002419, A274998, A279219, A305653, A317017, A317019, A317020.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(add(
      (3*d-1)*binomial(d+2, 3)/2*d, d=numtheory
      [divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 19 2018
  • Mathematica
    nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^((3 k - 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 26; CoefficientList[Series[Exp[Sum[x^k (1 + 5 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d - 1)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002419(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 5*x^k)/(k*(1 - x^k)^5)).
a(n) ~ 1/(2^(1987/2160) * 3^(713/1080) * 7^(173/2160) * n^(1253/2160) * Pi^(7/360)) * exp(-1/72 + (1/12-Zeta'(-1))/6 - Zeta(3)/(30 * Pi^2) + (111 * Zeta(5))/(200 * Pi^4) - (7056 * Zeta(3) * Zeta(5)^2)/Pi^12 - (592704 * Zeta(5)^3)/(5 * Pi^14) + (43016085504 * Zeta(5)^5)/(5 * Pi^24) + (2 * Zeta'(-3))/3 + ((-7 * (7/2)^(1/6) * Pi)/(3200 * 3^(2/3)) + (14 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/Pi^7 + (1029 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(5 * Pi^9) - (17978688 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/Pi^19) * n^(1/6) + (-((7/6)^(1/3) * Zeta(3))/(2 * Pi^2) - (7 * 3^(2/3) * (7/2)^(1/3) * Zeta(5))/(5 * Pi^4) + (75264 * 6^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/2) * Pi)/60 - (1008 * sqrt(14) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((6 * 6^(1/3) * 7^(2/3) * Zeta(5))/Pi^4) * n^(2/3) + ((2 * (2/7)^(1/6) * 3^(2/3) * Pi)/5) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 141, 86, 22, 1, 1, 1, 9, 36, 105, 216, 331, 310, 160, 30, 1, 1, 1, 10, 45, 148, 357, 672, 855, 692, 282, 42, 1, 1, 1, 11, 55, 201, 554, 1232, 1982, 2214, 1483, 500, 56, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 11 2017

Keywords

Comments

A(n,k) is the Euler transform of j -> binomial(j+k-2,k-1) evaluated at n.

Examples

			Square array begins:
1,  1,   1,   1,    1,    1,  ...
1,  1,   1,   1,    1,    1,  ...
1,  2,   3,   4,    5,    6,  ...
1,  3,   6,  10,   15,   21,  ...
1,  5,  13,  26,   45,   71,  ...
1,  7,  24,  59,  120,  216,  ...

Links

Crossrefs

Columns k=0..8 give A000012, A000041, A000219, A000294, A000335, A000391, A000417, A000428, A255965.
Main diagonal gives A293554.
Cf. A007318, A096751 (a similar but different sequence).

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-2, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 17 2017
  • Mathematica
    Table[Function[k, SeriesCoefficient[E^(Sum[x^i/(i (1 - x^i)^k), {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

Formula

G.f. of column k: exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).
For asymptotics of column k see comment from Vaclav Kotesovec in A255965.

A344099 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

Original entry on oeis.org

1, 1, 5, 20, 60, 190, 561, 1651, 4720, 13300, 36716, 99872, 267836, 708890, 1854255, 4796273, 12279445, 31135188, 78236006, 194921680, 481758832, 1181675902, 2877646681, 6959866116, 16723591530, 39934902812, 94795718409, 223741936855, 525206126933, 1226393510220
Offset: 0

Views

Author

Ilya Gutkovskiy, May 09 2021

Keywords

Crossrefs

Cf. A000332, A000391, A028377, A258343, A305206, A344100, A344101.

Programs

  • Mathematica
    nmax = 29; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 4], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 29}]

Formula

G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^5) ).
Previous Showing 11-16 of 16 results.

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