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 11-18 of 18 results.

A341366 Expansion of (1 / theta_4(x) - 1)^5 / 32.

Original entry on oeis.org

1, 10, 60, 275, 1060, 3612, 11210, 32310, 87665, 226130, 558684, 1329720, 3062905, 6853310, 14941330, 31820642, 66343150, 135659570, 272496680, 538427720, 1047788137, 2010303890, 3806292130, 7118038360, 13157217715, 24055170690, 43527162380, 77994164515, 138463246700
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Crossrefs

Cf. A002448, A004406, A014968, A015128, A327383, A338223, A340481, A341223, A341364, A341365, A341367, A341368, A341369, A341370.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..33);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^5/32, {x, 0, nmax}], x] // Drop[#, 5] &
nmax = 33; CoefficientList[Series[(1/32) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

Formula

G.f.: (1/32) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^5.

A341367 Expansion of (1 / theta_4(x) - 1)^6 / 64.

Original entry on oeis.org

1, 12, 84, 442, 1932, 7392, 25551, 81468, 243126, 686400, 1848156, 4775874, 11904215, 28737732, 67416756, 154122912, 344177823, 752310720, 1612395007, 3393652848, 7023685794, 14311193104, 28737793986, 56924936052, 111323290934, 215095157964, 410895944148, 776529566516
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Crossrefs

Cf. A002448, A004407, A014968, A015128, A327384, A338223, A340905, A341225, A341364, A341365, A341366, A341368, A341369, A341370.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..33);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^6/64, {x, 0, nmax}], x] // Drop[#, 6] &
nmax = 33; CoefficientList[Series[(1/64) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

Formula

G.f.: (1/64) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^6.

A160549 Omit first term from A160539 and divide by 7.

Original entry on oeis.org

0, 1, 5, 20, 70, 221, 646, 1772, 4614, 11490, 27537, 63808, 143514, 314279, 671872, 1405260, 2881030, 5799093, 11476452, 22357584, 42922558, 81284699, 151974124, 280739800, 512761178, 926568075, 1657448779, 2936506316, 5155349836, 8972488674, 15487146900
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Comments

These are Watson's coefficients beta'_n on page 125.

Examples

			G.f. = x + 5*x^2 + 20*x^3 + 70*x^4 + 221*x^5 + 646*x^6 + ...

Links

Crossrefs

Cf. A160539.
Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), A277968 (k=3), A277974 (k=5), this sequence (k=7), A277912 (k=11).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(Product[(1 - x^(7*j))/(1 - x^j)^7, {j, 1, nmax}] - 1)/7, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
  • PARI
    x='x+O('x^66); concat([0],Vec(eta(x^7)/eta(x)^7-1)/7) \\ Joerg Arndt, Nov 27 2016

Formula

From Seiichi Manyama, Nov 07 2016: (Start)
a(n) = A160539(n)/7, n > 0.
G.f.: ((Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^7) - 1)/7. (End)
a(n) ~ 2^(5/4) * exp(4*Pi*sqrt(2*n/7)) / (7^(13/4) * n^(9/4)). - Vaclav Kotesovec, Nov 10 2016

Extensions

Typo in definition corrected by Seiichi Manyama, Nov 07 2016

A277912 Expansion of ((Product_{n>=1} (1 - x^(11*n))/(1 - x^n)^11) - 1)/11 in powers of x.

Original entry on oeis.org

0, 1, 7, 38, 175, 714, 2653, 9139, 29563, 90650, 265401, 746142, 2023566, 5314008, 13554912, 33673525, 81654104, 193646588, 449903128, 1025532912, 2296519589, 5058078488, 10968488747, 23440057192, 49406752403, 102792264765, 211242738976, 429066735314, 861868377262, 1713014236294, 3370525567099
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f. = x + 7*x^2 + 38*x^3 + 175*x^4 + 714*x^5 + 2653*x^6 + ...

Links

Crossrefs

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), A277968 (k=3), A277974 (k=5), A160549 (k=7), this sequence (k=11).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(Product[(1 - x^(11*j))/(1 - x^j)^11, {j, 1, nmax}] - 1)/11, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^11] / QPochhammer[ x]^11 - 1) / 11, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^11 + A) / eta(x + A)^11 - 1) / 11, n))}; /* Michael Somos, Nov 13 2016 */
  • PARI
    x='x+O('x^66); concat([0],Vec(eta(x^11)/eta(x)^11-1)/11) \\ Joerg Arndt, Nov 27 2016

Formula

G.f.: ((Product_{n>=1} (1 - x^(11*n))/(1 - x^n)^11) - 1)/11.
a(n) ~ 5^(11/4) * exp(4*Pi*sqrt(5*n/11)) / (sqrt(2)*11^(17/4)*n^(13/4)). - Vaclav Kotesovec, Nov 10 2016

A277968 Expansion of ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3 in powers of x.

Original entry on oeis.org

0, 1, 3, 7, 16, 33, 66, 125, 231, 412, 720, 1227, 2056, 3380, 5478, 8745, 13792, 21483, 33114, 50510, 76344, 114356, 169920, 250503, 366666, 532975, 769758, 1104847, 1576640, 2237331, 3158208, 4435502, 6199479, 8624820, 11946096, 16475880, 22630864, 30962990
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f. = x + 3*x^2 + 7*x^3 + 16*x^4 + 33*x^5 + 66*x^6 + ...

Links

Crossrefs

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), this sequence (k=3), A277974 (k=5), A160549 (k=7), A277912 (k=11).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(Product[(1 - x^(3*j))/(1 - x^j)^3, {j, 1, nmax}] - 1)/3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / QPochhammer[ x]^3 - 1) / 3, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)
  • PARI
    first(n)=my(x='x); concat([0], Vec((prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))-1)/3)) \\ Charles R Greathouse IV, Nov 07 2016
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A)^3 - 1) / 3, n))}; /* Michael Somos, Nov 13 2016 */

Formula

a(n) = A273845(n)/3, n > 0.
G.f.: ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (27*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016

A341371 Expansion of (1 / theta_4(x) - 1)^10 / 1024.

Original entry on oeis.org

1, 20, 220, 1750, 11220, 61424, 297485, 1305260, 5276930, 19905700, 70742012, 238662710, 769055130, 2378885080, 7093202060, 20459149350, 57254003225, 155851688980, 413590326020, 1072076963640, 2719067915088, 6757856447720, 16480738170760, 39486206985530, 93043172921735
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A002448, A004411, A014968, A015128, A327388, A338223, A340947, A341236, A341364, A341365, A341366, A341367, A341368, A341369, A341370.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..34);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 34; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^10/1024, {x, 0, nmax}], x] // Drop[#, 10] &
nmax = 34; CoefficientList[Series[(1/1024) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: (1/1024) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^10.

A277992 b(n, 2) where b(n, m) is defined by expansion of ((Product_{k>=1} (1 - x^(prime(n)*k))/(1 - x^k)^prime(n)) - 1)/prime(n) in powers of x.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157
Offset: 1

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Comments

c(n, m) is defined by expansion of (Product_{k>=1} 1/(1 - x^k)^prime(n))/prime(n) in powers of x.
b(n, 2) = c(n, 2) for n > 1.

Examples

			a(1) = b(1, 2) = A014968(2) = 2.
a(2) = b(2, 2) = A277968(2) = c(2, 2) = A000716(2)/3 = 3.
a(3) = b(3, 2) = A277974(2) = c(3, 2) = A023004(2)/5 = 4.
a(4) = b(4, 2) = A160549(2) = c(4, 2) = A023006(2)/7 = 5.
a(5) = b(5, 2) = A277912(2) = c(5, 2) = A023010(2)/11 = 7.

Crossrefs

Cf. A014968, A277968, A277974, A160549, A277912.
Expansion of Product_{k>=1} 1/(1 - x^k)^prime(n): A000712 (n=1), A000716 (n=2), A023004 (n=3), A023006 (n=4), A023010 (n=5).

Formula

a(n) = A098090(n - 1) = (prime(n) + 3)/2 for n > 1.

A359481 Irregular triangle read by rows in which T(n,k) is one half of the number of overpartitions of n having k distinct parts, n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 2, 10, 4, 12, 4, 2, 22, 8, 4, 26, 20, 3, 34, 40, 4, 44, 60, 8, 2, 54, 100, 16, 6, 58, 148, 40, 2, 74, 208, 80, 4, 88, 268, 160, 4, 88, 388, 240, 16, 5, 110, 468, 416, 32, 2, 118, 616, 616, 80, 6, 136, 736, 936, 160, 2, 142, 940, 1296, 320, 6, 162, 1108, 1816, 576
Offset: 1

Views

Author

Omar E. Pol, Mar 31 2023

Keywords

Examples

			Triangle begins:
  1;
  2;
  2,  2;
  3,  4;
  2, 10;
  4, 12,   4;
  2, 22,   8;
  4, 26,  20;
  3, 34,  40;
  4, 44,  60,   8;
  2, 54, 100,  16;
  6, 58, 148,  40;
  2, 74, 208,  80;
  4, 88, 268, 160;
  4, 88, 388, 240, 16;
  ...

Crossrefs

Column 1 gives A000005.
Row n has length A003056(n).
Row sums give A014968, n >= 1.
The first element of column k is A000079(k-1).
The first element of column k is in row A000217(k).
Cf. A116608, A235790.

Formula

T(n,k) = (1/2)*A235790(n,k).
T(n,k) = (2^(k-1))*A116608(n,k).
Previous Showing 11-18 of 18 results.

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

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