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 1-5 of 5 results.

A160525 Coefficients in the expansion of C/B^2, in Watson's notation of page 118.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 109, 183, 295, 471, 732, 1129, 1705, 2554, 3769, 5517, 7979, 11458, 16289, 23007, 32227, 44869, 62028, 85284, 116530, 158432, 214228, 288348, 386224, 515156, 684109, 904963, 1192353, 1565383, 2047642, 2669591, 3468797, 4493351, 5802533
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 109*x^7 + ...
G.f. = q^5 + 2*q^29 + 5*q^53 + 10*q^77 + 20*q^101 + 36*q^125 + 65*q^149 + 109*q^173 + ...

Links

Crossrefs

Cf. A160526, A160527, A160528.
Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), A278690 (k=3), A160461 (k=5), this sequence (k=7).

Programs

  • Maple
    M1:=1200:
fm:=mul(1-x^n,n=1..M1):
A:=x^(1/7)*subs(x=x^(24/7),fm):
B:=x*subs(x=x^24,fm):
C:=x^7*subs(x=x^168,fm):
t1:=C/B^2;
t2:=series(t1,x,M1);
t3:=subs(x=y^(1/24),t2/x^5);
t4:=series(t3,y,M1/24);
t5:=seriestolist(t4); # A160525
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 13 2017 *)

Formula

See Maple code for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^2. - Seiichi Manyama, Nov 06 2016
a(n) ~ sqrt(13/3) * exp(sqrt(26*n/21)*Pi) / (28*n). - Vaclav Kotesovec, Apr 13 2017

A160528 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 118.

Original entry on oeis.org

1, 5, 20, 65, 190, 506, 1265, 2986, 6745, 14645, 30767, 62745, 124706, 242110, 460337, 858673, 1574140, 2839862, 5048435, 8852562, 15327290, 26224173, 44372688, 74301095, 123200079, 202394897, 329596348, 532299955, 852914900, 1356426196, 2141819621
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			G.f. = 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 506*x^5 + 1265*x^6 + ...
G.f. = q^23 + 5*q^47 + 20*q^71 + 65*q^95 + 190*q^119 + 506*q^143 + 1265*q^167 + ...

Links

Crossrefs

Cf. A002300, A160525, A160526, A160527.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))^4 /(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

Formula

See Maple code in A160525 for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))^4/(1 - x^n)^5. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(62*n/21)) * sqrt(31) / (4*sqrt(3) * 7^(5/2) * n). - Vaclav Kotesovec, Nov 10 2017

A160526 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 118.

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 221, 427, 804, 1461, 2596, 4497, 7652, 12767, 20984, 33958, 54255, 85580, 133520, 206066, 315010, 477083, 716494, 1067316, 1578102, 2316569, 3377965, 4894045, 7047970, 10091120, 14369439, 20354090, 28687663, 40239129, 56183879
Offset: 0

Views

Author

N. J. A. Sloane, Nov 13 2009

Keywords

Examples

			G.f. = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + ...
G.f. = q^11 + 3*q^35 + 9*q^59 + 22*q^83 + 51*q^107 + 108*q^131 + 221*q^155 + ...

Links

Crossrefs

Cf. A160525, A160527, A160528.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))^2 /(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

Formula

See Maple code in A160525 for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))^2/(1 - x^n)^3. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(38*n/21)) * sqrt(19) / (4*sqrt(3) * 7^(3/2) * n). - Vaclav Kotesovec, Nov 10 2017

A277958 Expansion of Product_{n>=1} (1 - x^(7*n))^7/(1 - x^n)^8 in powers of x.

Original entry on oeis.org

1, 8, 44, 192, 726, 2464, 7704, 22521, 62281, 164252, 415796, 1015334, 2401462, 5519640, 12363062, 27047913, 57917068, 121588588, 250638216, 507974950, 1013409244, 1992161935, 3862461694, 7392045512, 13975011909, 26116935550, 48277368020, 88320521108, 159993054081
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2016

Keywords

Examples

			G.f.: 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + 7704*x^6 + ...

Links

Crossrefs

Cf. A071746, A160527, A213261.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^7 /(1 - x^k)^8 , {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^7/(1 - x^n)^8.
A213261(n) = 7*A160527(n) + 49*a(n - 1) for n > 0 due to Ramanujan's congruences.
a(n) ~ exp(Pi*sqrt(98*n/21)) / (1372*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017

A160533 Coefficients in the expansion of C^5/B^6, in Watson's notation of page 118.

Original entry on oeis.org

1, 6, 27, 98, 315, 918, 2492, 6367, 15495, 36145, 81326, 177219, 375461, 775544, 1565870, 3096615, 6008917, 11458720, 21502964, 39754385, 72485518, 130464603, 231989748, 407847488, 709365160, 1221364655, 2082872680, 3519963776, 5897536697, 9800358525
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Examples

			G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + ...
G.f. = q^29 + 6*q^53 + 27*q^77 + 98*q^101 + 315*q^125 + 918*q^149 + 2492*q^173 + ...

Links

Crossrefs

Cf. A160525, A160526, A160527, A160528.

Programs

  • Mathematica
    nn = 29; CoefficientList[Series[Product[(1 - x^(7 n))^5/(1 - x^n)^6, {n, nn}], {x, 0, nn}], x] (* Michael De Vlieger, Nov 06 2016 *)

Formula

See Maple code in A160525 for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))^5/(1 - x^n)^6. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(74*n/21)) * sqrt(37) / (1372*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.