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.

A282919 a(n) = A000041(49*n + 47).

Original entry on oeis.org

124754, 118114304, 24908858009, 2366022741845, 133978259344888, 5234371069753672, 154043597379576030, 3617712763867604423, 70593393646562135510, 1178875491155735802646, 17229817230617210720599, 224282898599046831034631, 2636785814481962651219075
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 179.

Links

Crossrefs

Cf. A160528, A282920, A282921, A282922, A282923, A282924, A282925, A282926, A282927, A282928, A282929, A282930, A282931, A282932.
Cf. A000041, A213261 (p(7*n + 5)), A277958, A278559 (p(25*n + 24)), this sequence (p(49*n + 47)).

Programs

  • Mathematica
    Table[PartitionsP[49n+47],{n,0,12}] (* Indranil Ghosh, Feb 25 2017 *)
  • PARI
    a(n) = numbpart(49*n+47); \\ Indranil Ghosh, Feb 25 2017

Formula

a(n) = A213261(7*n + 6) = A000041(49*n + 47).
a(n) = 2546 * 7^2 * A160528(n) + 48934 * 7^4 * A282920(n-1) + 1418989 * 7^5 * A282921(n-2) + 2488800 * 7^7 * A282922(n-3) + 2394438 * 7^9 * A282923(n-4) + 1437047 * 7^11 * A282924(n-5) + 4043313 * 7^12 * A282925(n-6) + 161744 * 7^15 * A282926(n-7) + 32136 * 7^17 * A282927(n-8) + 31734 * 7^18 * A282928(n-9) + 3120 * 7^20 * A282929(n-10) + 204 * 7^22 * A282930(n-11) + 8 * 7^24 * A282931(n-12) + 7^25 * A282932(n-13) for n >= 13.

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

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

A002300 Coefficients in the expansion of B^2*C^3 in Watson's notation of page 118.

Original entry on oeis.org

1, -2, -1, 2, 1, 2, -2, -3, 4, 1, -5, -3, -6, 8, 3, 4, 8, -3, 0, -2, -8, -4, -4, -13, 9, 5, 18, -2, -2, -8, -3, 10, 0, -4, 2, 19, -14, 7, -8, 0, -20, -4, -1, 8, -2, -15, -7, 8, 26, -10, 26, 18, 10, -2, 10, -28, -29, 18, -20, -15, 6, -8, 8, -8, 2, 19, -1, 0, -8, -6, 28, -26, -6, 23, -1, 4, 12, 25, -36, -14, 8, 0, 18, 20, 21, -12, -3, -9, 0, -16, -48
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Although Watson says these are the coefficients theta_n defined on page 128, it appears that this is a mistake, and they are really the coefficients theta'_n. The true theta_n are given in A160528.
Watson's main reason for computing this sequence was to study values of n such that partition(49n+47) == 0 mod 343 (cf. A160553).

Examples

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^5 - 2*x^6 - 3*x^7 + 4*x^8 + x^9 - 5*x^10 + ...
G.f. = q^23 - 2*q^47 - q^71 + 2*q^95 + q^119 + 2*q^143 - 2*q^167 - 3*q^191 + 4*q^215 + ...

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A160553.

Programs

  • Maple
    M1:=2400:
fm:=mul(1-x^n,n=1..M1):
B:=x*subs(x=x^24,fm):
C:=x^7*subs(x=x^168,fm):
t1:=B^2*C^3;
t2:=series(t1,x,M1);
t3:=subs(x=y^(1/24),t2/x^23);
t4:=series(t3,y,M1/24);
t5:=seriestolist(t4); # A002300
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^7]^3, {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^7 + A)^3, n))}; /* Michael Somos, May 31 2012 */

Formula

Expansion of q^(-23/24) * eta(q)^2 * eta(q^7)^3 in powers of q. - Michael Somos, May 31 2012
Euler transform of period 7 sequence [ -2, -2, -2, -2, -2, -2, -5, ...]. - Michael Somos, May 31 2012
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(7*k))^3. - Michael Somos, May 31 2012

Extensions

Entry revised by N. J. A. Sloane, Nov 14 2009

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.

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