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 1-7 of 7 results.

A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8.

Original entry on oeis.org

1, 6, 24, 77, 216, 552, 1315, 2964, 6387, 13255, 26640, 52074, 99336, 185430, 339483, 610709, 1081227, 1886484, 3247502, 5521365, 9279624, 15429149, 25397088, 41412030, 66928700, 107265576, 170556654, 269164346, 421765920, 656419080, 1015044526, 1559950185, 2383284894
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A002448, A004404, A014968, A015128, A063691, A319552, A327381, A338223, A341221, 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, 3):
seq(a(n), n=3..35);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 35; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^3/8, {x, 0, nmax}], x] // Drop[#, 3] &
nmax = 35; CoefficientList[Series[(1/8) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

Formula

G.f.: (1/8) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^3.
a(n) ~ A319552(n)/8 ~ 3*exp(Pi*sqrt(3*n)) / (512*n^(3/2)). - Vaclav Kotesovec, Feb 20 2021

A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.

Original entry on oeis.org

1, 8, 40, 156, 520, 1552, 4262, 10960, 26716, 62276, 139744, 303412, 640001, 1315832, 2644004, 5204044, 10052182, 19086348, 35672516, 65708116, 119409576, 214289116, 380068582, 666723748, 1157550524, 1990230968, 3390558072, 5726064688, 9590759624, 15938198484, 26289242026
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A002448, A004405, A014968, A015128, A063730, A284286, A327382, A338223, A341222, A341364, 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, 4):
seq(a(n), n=4..34);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 34; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^4/16, {x, 0, nmax}], x] // Drop[#, 4] &
nmax = 34; CoefficientList[Series[(1/16) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: (1/16) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^4.
a(n) ~ A284286(n)/16. - Vaclav Kotesovec, Feb 20 2021

A341368 Expansion of (1 / theta_4(x) - 1)^7 / 128.

Original entry on oeis.org

1, 14, 112, 665, 3248, 13776, 52437, 183080, 595399, 1824109, 5310144, 14787542, 39605363, 102465972, 257005641, 626841236, 1490521109, 3462881324, 7875519169, 17562223791, 38456245849, 82793422502, 175452110162, 366348547908, 754392685046, 1533283745644, 3078157040665
Offset: 7

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A002448, A004408, A014968, A015128, A327385, A338223, A340906, A341226, A341364, A341365, A341366, A341367, 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, 7):
seq(a(n), n=7..33);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^7/128, {x, 0, nmax}], x] // Drop[#, 7] &
nmax = 33; CoefficientList[Series[(1/128) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &

Formula

G.f.: (1/128) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^7.

A341369 Expansion of (1 / theta_4(x) - 1)^8 / 256.

Original entry on oeis.org

1, 16, 144, 952, 5136, 23904, 99292, 376512, 1324376, 4372632, 13673888, 40787848, 116713350, 321861312, 858693192, 2223428224, 5602833292, 13772292360, 33089930724, 77846837848, 179602530648, 406914172336, 906438716196, 1987418937952, 4293164981849, 9144987747024
Offset: 8

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A002448, A004409, A014968, A015128, A319553, A327386, A338223, A340915, A341227, A341364, A341365, A341366, A341367, A341368, 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, 8):
seq(a(n), n=8..33);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^8/256, {x, 0, nmax}], x] // Drop[#, 8] &
nmax = 33; CoefficientList[Series[(1/256) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

Formula

G.f.: (1/256) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^8.

A341370 Expansion of (1 / theta_4(x) - 1)^9 / 512.

Original entry on oeis.org

1, 18, 180, 1311, 7740, 39204, 176388, 721530, 2728053, 9651056, 32246892, 102515508, 311923386, 912771468, 2579132196, 7060677537, 18781247700, 48660380190, 123061973176, 304351869708, 737293187286, 1752035386188, 4089222211212, 9384936015492, 21201250825554
Offset: 9

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A002448, A004410, A014968, A015128, A327387, A338223, A340946, A341228, A341364, A341365, A341366, A341367, A341368, A341369.

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, 9):
seq(a(n), n=9..33);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^9/512, {x, 0, nmax}], x] // Drop[#, 9] &
nmax = 33; CoefficientList[Series[(1/512) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

Formula

G.f.: (1/512) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^9.

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.

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.
Showing 1-7 of 7 results.

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