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.

A364736 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 0, -3, -3, 17, 45, -90, -546, 130, 5832, 7074, -53625, -159214, 374517, 2419131, -728364, -30011530, -37519884, 307731042, 940757526, -2343385995, -15421126275, 5164279686, 203045257272, 255851517115, -2186669342070, -6760669947375, 17391580425180
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A364735, A364737, A364738.
Cf. A300048.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(n+2*k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(n+2*k,n-1-k) for n > 0.

A364735 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 0, -2, -1, 8, 10, -37, -84, 168, 660, -624, -4950, 583, 35464, 23166, -240513, -359008, 1511640, 3898100, -8387664, -36522256, 35444728, 311764768, -25659766, -2466384737, -1793133360, 18077558170, 28951038285, -120750295320, -330486900870
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A364736, A364737, A364738.
Cf. A106228.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(n+k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(n+k,n-1-k) for n > 0.

A364737 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 0, -4, -6, 28, 119, -116, -1820, -2128, 22212, 79877, -172700, -1652728, -857428, 25387284, 71506309, -268817888, -1838702048, 449975584, 33164610276, 68575577309, -429542625096, -2221814345660, 2539462697398, 46048818685880, 61721413191310
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A364735, A364736, A364738.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(n+3*k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(n+3*k,n-1-k) for n > 0.

A365088 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x))^5.

Original entry on oeis.org

1, 1, -4, 1, 46, -129, -405, 3319, -1617, -59258, 199541, 642170, -6038395, 3886091, 119884973, -440626784, -1367688245, 14055527190, -11043763380, -290488387366, 1137260033731, 3336325340735, -36966844508130, 34098313310315, 776097820004580
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Crossrefs

Cf. A090192, A365085, A365086, A365087.
Cf. A321799, A364738, A365084.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(n+4*k-1, n-k)/(n-k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(n+4*k-1,n-k) / (n-k+1).

A363782 Products of three distinct strong primes.

Original entry on oeis.org

5423, 6919, 7667, 11033, 11803, 12529, 13079, 13277, 14773, 16687, 18139, 18241, 18821, 18887, 20009, 20213, 21373, 22649, 23749, 24013, 25201, 25619, 25789, 26609, 27269, 27863, 28897, 29087, 30217, 30481, 30943, 32021, 32153, 32219, 33031, 33473, 34133, 35003, 35629, 35717, 36839
Offset: 1

Views

Author

Massimo Kofler, Jun 21 2023

Keywords

Comments

Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2.

Examples

			5423 = 11*17*29 and 11 > (7+13)/2, 17 > (13+19)/2, 29 > (23+31)/2.
6919 = 11*17*37 and 11 > (7+13)/2, 17 > (13+19)/2, 37 > (31+41)/2.

Links

Crossrefs

Cf. A007304, A051634, A363167, A364738.

Programs

  • Mathematica
    strongQ[p_] := p > 2 && 2*p > Total[NextPrime[p, {-1, 1}]]; Select[Range[1, 37000, 2], (f = FactorInteger[#])[[;; , 2]] == {1, 1, 1} && AllTrue[f[[;; , 1]], strongQ] &] (* Amiram Eldar, Jun 21 2023 *)
Module[{nn=50,strgpr},strgpr=Select[Partition[Prime[Range[nn]],3,1],#[[2]]>(#[[1]]+#[[3]])/2&][[;;,2]];Take[Union[Times@@@Subsets[strgpr,{3}]],nn]] (* Harvey P. Dale, Aug 21 2024 *)

Extensions

Definition clarified by N. J. A. Sloane, Oct 08 2023
Showing 1-5 of 5 results.

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

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