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 1-8 of 8 results.

A305882 -1 + Product_{n>=1} 1/(1 + a(n)*x^n) = g.f. of A000040 (prime numbers).

Original entry on oeis.org

-2, 1, 1, 4, 4, 13, 16, 44, 52, 112, 182, 411, 620, 1318, 2142, 5148, 7676, 15228, 27530, 58660, 98372, 207392, 364464, 763263, 1341508, 2773990, 4923220, 10470948, 18510902, 37546152, 69269976, 148419094, 258284232, 534761242, 981480012, 2004302204
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 13 2018

Keywords

Examples

			1/((1 - 2*x) * (1 + x^2) * (1 + x^3) * (1 + 4*x^4) * (1 + 4*x^5) * ... * (1 + a(n)*x^n) * ...) =  1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + ... + A000040(k)*x^k + ...

Crossrefs

Cf. A000040, A030010, A030018, A145519, A147541, A147557, A305871, A305881.

Formula

Product_{n>=1} 1/(1 + a(n)*x^n) = 1 + Sum_{k>=1} prime(k)*x^k.
Product_{n>=1} (1 + a(n)*x^n) = Sum_{k>=0} A030018(k)*x^k.

A348128 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, -1, -1, 2, 1, 0, -2, 0, 1, 3, -2, -1, 0, 4, 0, -1, -4, 6, 2, 2, -10, 4, 4, 13, -15, -7, -2, 30, -7, -7, -33, 42, 8, 16, -70, 27, 22, 95, -116, -21, -39, 223, -61, -48, -261, 326, 51, 129, -581, 242, 109, 752, -932, -105, -330, 1806, -612, -240, -2140, 2750, 227, 1245, -4865
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 01 2021

Keywords

Links

Crossrefs

Cf. A000040, A002121, A010051, A030010, A147557, A305871, A305882, A308298, A328777, A348127.

A353605 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} prime(n+1)*x^n.

Original entry on oeis.org

3, 5, -8, 35, -52, 118, -320, 1597, -2016, 6616, -16064, 40516, -122552, 381606, -903176, 4389807, -7597004, 22835416, -61172890, 188526110, -486889660, 1550995910, -4093173788, 11608277912, -33815484714, 105179650108, -279683446078, 883705997682, -2366564864546
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2022

Keywords

Crossrefs

Cf. A006005, A065091, A147557, A353606.

Programs

  • Mathematica
    nn = 29; f[x_] := Product[(1 + a[n] x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sum[Prime[k + 1] x^k, {k, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A353606 Product_{n>=1} (1 + a(n)*x^n) = 1 + x + Sum_{n>=2} prime(n-1)*x^n.

Original entry on oeis.org

1, 2, 1, 4, 1, 0, -1, 11, -3, -7, 4, -10, 5, -15, -4, 151, -9, -50, -3, -63, 19, -176, 6, 591, -27, -637, 13, -999, 50, -1957, -49, 27250, -162, -7887, 83, -12821, 468, -27226, -40, 127341, -1215, -99166, -526, -174140, 2640, -362870, 1673, 1419061, -4516, -1344620
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2022

Keywords

Crossrefs

Cf. A008578, A147557, A353605.

Programs

  • Mathematica
    nn = 50; f[x_] := Product[(1 + a[n] x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - x - Sum[Prime[k - 1] x^k, {k, 2, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A348127 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

Original entry on oeis.org

0, 1, 1, -1, 0, -1, 1, -1, -1, -1, 2, 0, 0, -3, 0, 0, 3, -3, -1, -1, 4, -4, -1, -5, 6, 2, 2, -17, 4, 4, 13, -16, -7, -11, 30, -14, -7, -34, 42, 7, 16, -80, 27, 6, 95, -117, -21, -60, 223, -97, -48, -265, 326, 53, 129, -800, 242, 93, 752, -948, -105, -499, 1806, -853, -240, -2189, 2750, 124
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 01 2021

Keywords

Links

Crossrefs

Cf. A000040, A002121, A010051, A030010, A147557, A305871, A305882, A308298, A328777, A348128.

Programs

  • Maple
    N:= 20: # for a(1)..a(N)
P:= 1: a:= Vector(N):
for n from 1 to N do
  c:= coeff(P,x,n);
  if isprime(n) then a[n]:= 1-c  else a[n]:= -c fi;
  P:= series(P/(1-a[n]*x^n),x,N+1);
od:
convert(a,list); # Robert Israel, Mar 01 2022

A353950 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} prime(n+1)*x^n.

Original entry on oeis.org

3, -4, -8, -26, -52, -126, -320, -1214, -2016, -7068, -16064, -48142, -122552, -390574, -903176, -3549556, -7597004, -22902332, -61172890, -198872948, -486889660, -1555059566, -4093173788, -12448334478, -33815484714, -105268420776, -279683446078, -894795490384, -2366564864546
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2022

Keywords

Crossrefs

Cf. A006005, A065091, A147557, A305882, A353605, A353606, A353951.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, Prime[n + 1], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 29]

A353951 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + x + Sum_{n>=2} prime(n-1)*x^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, -1, -2, -3, -2, 4, -1, 5, 2, -4, -8, -9, -3, -3, 12, 19, -6, 6, -38, -27, -32, 13, 56, 50, 99, -49, -135, -162, -258, 83, 114, 468, 359, -40, -390, -1215, -791, -526, 876, 2640, 1816, 1673, -3404, -4516, -6527, -3640, 5320, 9282, 18019, 7210
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2022

Keywords

Crossrefs

Cf. A008578, A147557, A305882, A353605, A353606, A353950.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, If[n == 1, 1, Prime[n - 1]], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 55]

A152006 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

Original entry on oeis.org

1, 1, 2, 5, 8, 18, 34, 63, 102, 203, 336, 589, 999, 1675, 2799, 4768, 7561, 12224, 20513, 31724, 51621, 81976, 128560, 199192, 312536, 482806, 744847, 1147952, 1755931, 2649474, 4051413, 6069450, 9105323, 13747364, 20335077, 30508629, 45198631
Offset: 0

Views

Author

Roger L. Bagula, Nov 19 2008

Keywords

Crossrefs

Cf. A147557, A147654, A147655, A147665, A147869, A147871, A147880, A147953.

Programs

  • Mathematica
    f[n_] = If[n < 2, n, Prime[n - 1]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 37], x],37]
(* Program edited and corrected by Petros Hadjicostas, Apr 12 2020 *)

Formula

a(n) = [x^n] Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

Extensions

Various sections edited by Petros Hadjicostas, Apr 12 2020
Showing 1-8 of 8 results.

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