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.

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A294036 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], 1).

Original entry on oeis.org

1, 4, 16, 64, 280, 1504, 9856, 70144, 498136, 3449440, 23506816, 160566784, 1115048896, 7905796864, 56994288640, 414928113664, 3034880623576, 22255957312864, 163667338903936, 1208070406612480, 8955840250934080, 66678657938510080
Offset: 0

Views

Author

Peter Luschny, Nov 02 2017

Keywords

Comments

Diagonal of rational function 1/(1 - (x^4 + y^4 + z^4 + t^4 + 4*x*y*z*t)). - Gheorghe Coserea, Aug 04 2018

Crossrefs

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = this seq., H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = A294037(n).

Programs

  • Maple
    T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1, k=0..m-2)], x):
lprint(seq(simplify(T(4,n,1)), n=0..39));
  • Mathematica
    Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Nov 02 2017 *)

Formula

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(4, n, 1).
a(n) ~ 2^(3*n + 2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Nov 02 2017

A294037 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], -1).

Original entry on oeis.org

1, 4, 16, 64, 232, 544, -1664, -37376, -362024, -2743712, -17780864, -98955776, -442825664, -1129423616, 5536033792, 118591811584, 1224814969816, 9905491019104, 68032143081856, 398051159254528, 1854461906222272, 4784426026102528
Offset: 0

Views

Author

Peter Luschny, Nov 02 2017

Keywords

Comments

Diagonal of rational function 1/(1 - (x^4 + y^4 + z^4 - t^4 + 4*x*y*z*t)). - Gheorghe Coserea, Aug 04 2018

Crossrefs

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = A294036(n), H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = this seq..

Programs

  • Maple
    T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1, k=0..m-2)], x):
lprint(seq(simplify(T(4,n,-1)), n=0..39));
  • Mathematica
    Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, -1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 02 2017 *)

Formula

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(4, n, -1).

A291898 (n+1)^2*a(n+1) = -(9*n^2 + 9*n + 3)*a(n) - 27*n^2*a(n-1), with a(0) = 1 and a(1) = -3.

Original entry on oeis.org

1, -3, 9, -21, 9, 297, -2421, 12933, -52407, 145293, -35091, -2954097, 25228971, -142080669, 602217261, -1724917221, 283305033, 38852066421, -337425235479, 1938308236731, -8364863310291, 24286959061533, -3011589296289, -574023003011199, 5028616107443691
Offset: 0

Views

Author

Michael Somos, Nov 02 2017

Keywords

Examples

			G.f. = 1 - 3*x + 9*x^2 - 21*x^3 + 9*x^4 + 297*x^5 - 2421*x^6 + ...

Links

Crossrefs

Cf. A005928, A006077, A121589.

Programs

  • Magma
    I:=[-3,9]; [1] cat [n le 2 select I[n] else (-1)*((9*n^2-9*n+3)*Self(n-1) + 27*(n-1)^2*Self(n-2))/n^2: n in [1..30]]; // G. C. Greubel, Jul 28 2018
  • Maple
    f:= gfun:-rectoproc({(n+1)^2*a(n+1) = -(9*n^2 + 9*n + 3)*a(n) - 27*n^2*a(n-1),a(0)=1,a(1)=-3},a(n),remember):
map(f, [$0..50]); # Robert Israel, Nov 02 2017
  • Mathematica
    a[ n_] := If[ n < 0, 0,(-3)^n HypergeometricPFQ[ {-n, 1 - n, 2 - n}/3, {1, 1}, 1]];
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1, 2}/3, {1}, (3 x / (1 + 3 x))^3 ] / (1 + 3 x), {x, 0, n}];
nxt[{n_,a_,b_}]:={n+1,b,(-(9 n^2+9n+3)b- a 27n^2)/(n+1)^2}; NestList[nxt,{1,1,-3},30][[;;,2]] (* Harvey P. Dale, Nov 02 2024 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( subst(eta(x + A)^3 / eta(x^3 + A), x, serreverse( x * eta(x^9 + A)^3 / eta(x + A)^3)), n))};

Formula

Given A(x) is the g.f. of this sequence, B(x) is the g.f. of A005928, and C(x) is the g.f. of A121589, then B(x) = A(C(x)).
a(n) = (-1)^n * A006077(n).
0 = y*(3 + 27*x) + y'*(1 + 18*x + 81*x^2) + y''*(x + 9*x^2 + 27*x^3) where y(x) is the g.f. of this sequence.

A291276 Primes p such that p does not divide any term of the Apery-like sequence A002893.

Original entry on oeis.org

2, 7, 13, 37, 61, 73, 109, 127, 157, 163, 193, 211, 223, 229, 271, 283, 307, 313, 331, 337, 349, 367, 379, 409, 421, 433, 463, 487, 499, 523, 577, 607, 613, 619, 631, 661, 673, 691, 727, 733, 751, 757, 769, 787, 823, 829, 853, 883, 907, 919, 1021, 1039
Offset: 1

Views

Author

N. J. A. Sloane, Aug 21 2017

Keywords

Links

Crossrefs

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291277 Primes p such that p does not divide any term of the Apery-like sequence A081085.

Original entry on oeis.org

3, 11, 17, 19, 43, 59, 73, 83, 89, 107, 179, 211, 227, 233, 241, 257, 307, 331, 337, 379, 401, 409, 419, 433, 449, 457, 467, 521, 547, 563, 577, 587, 593, 601, 619, 641, 643, 683, 691, 739, 761, 769, 811, 827, 859, 881, 883, 929, 937, 947, 953
Offset: 1

Views

Author

N. J. A. Sloane, Aug 21 2017

Keywords

Links

Crossrefs

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291279 Primes p such that p does not divide any term of the Apery-like sequence A093388.

Original entry on oeis.org

5, 11, 29, 31, 59, 79, 107, 131, 149, 151, 173, 179, 193, 197, 199, 241, 251, 271, 317, 409, 433, 439, 443, 457, 461, 509, 557, 587, 601, 607, 659, 677, 701, 727, 751, 769, 773, 797, 821, 823, 827, 919, 971, 1009, 1013, 1019, 1033, 1039, 1061, 1063, 1087
Offset: 1

Views

Author

N. J. A. Sloane, Aug 21 2017

Keywords

Links

Crossrefs

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291280 Primes p such that p does not divide any term of the Apery-like sequence A125143.

Original entry on oeis.org

2, 5, 7, 11, 13, 19, 29, 41, 47, 61, 67, 71, 73, 89, 97, 101, 103, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 223, 227, 229, 233, 241, 251, 257, 263, 269, 281, 283, 293, 317, 331, 347, 349, 359, 367, 373, 379, 383
Offset: 1

Views

Author

N. J. A. Sloane, Aug 21 2017

Keywords

Links

Crossrefs

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291281 Primes p such that p does not divide any term of the Apery-like sequence A229111.

Original entry on oeis.org

2, 3, 17, 19, 23, 31, 47, 53, 61, 107, 109, 113, 137, 139, 151, 173, 197, 199, 211, 227, 229, 233, 241, 257, 263, 293, 317, 347, 353, 383, 421, 439, 443, 467, 499, 541, 587, 593, 619, 647, 661, 677, 683, 691, 751, 769, 773, 857, 919
Offset: 1

Views

Author

N. J. A. Sloane, Aug 21 2017

Keywords

Links

Crossrefs

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291282 Primes p such that p does not divide any term of the Apery-like sequence A002895.

Original entry on oeis.org

3, 5, 13, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 83, 89, 101, 103, 107, 109, 127, 131, 137, 163, 167, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 257, 269, 307, 311, 317, 337, 347, 349, 359, 367, 373, 409, 419, 421, 449, 457
Offset: 1

Views

Author

N. J. A. Sloane, Aug 21 2017

Keywords

Links

Crossrefs

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291283 Primes p such that p does not divide any term of the Apery-like sequence A290575.

Original entry on oeis.org

3, 7, 13, 19, 23, 29, 31, 37, 43, 47, 61, 67, 73, 83, 89, 101, 103, 107, 109, 113, 127, 131, 139, 149, 157, 167, 179, 191, 193, 197, 223, 227, 229, 241, 251, 257, 271, 281, 293, 307, 311, 313, 353, 367, 373, 379, 389, 401, 419, 431, 433
Offset: 1

Views

Author

N. J. A. Sloane, Aug 21 2017

Keywords

Links

Crossrefs

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.
Previous Showing 41-50 of 50 results.

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

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