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.

A134759 a(n) = 2*A000984(n) - (n+1).

Original entry on oeis.org

1, 2, 9, 36, 135, 498, 1841, 6856, 25731, 97230, 369501, 1410852, 5408299, 20801186, 80233185, 310235024, 1202160763, 4667212422, 18150270581, 70690527580, 275693057619, 1076515748858, 4208197927417, 16466861455176, 64495207366175, 252821212875478
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Links

Crossrefs

Cf. A000108, A000984, A134758, A134760, A134761, A134762, A134763, A134770, A134771.

Programs

  • Magma
    [(n+1)*(2*Catalan(n)-1): n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[2 Binomial[2n,n]-n-1,{n,0,30}] (* Harvey P. Dale, Aug 07 2023 *)
  • SageMath
    [2*binomial(2*n,n) -(n+1) for n in range(41)] # G. C. Greubel, May 28 2024

Formula

From G. C. Greubel, May 28 2024: (Start)
a(n) = (n+1)*(2*A000108(n) - 1).
a(n) = (2*(2*n-1)*a(n-1) + 3*n*(n-1))/n.
G.f.: 2/sqrt(1-4*x) - 1/(1-x)^2.
E.g.f.: 2*exp(2*x)*BesselI(0, 2*x) - (1+x)*exp(x). (End)

Extensions

More terms from Harvey P. Dale, Aug 07 2023

A134762 a(n) = 3*A000984(n) - 2.

Original entry on oeis.org

1, 4, 16, 58, 208, 754, 2770, 10294, 38608, 145858, 554266, 2116294, 8112466, 31201798, 120349798, 465352558, 1803241168, 7000818658, 27225405898, 106035791398, 413539586458, 1614773623318, 6312296891158, 24700292182798, 96742811049298, 379231819313254
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Comments

Second inverse binomial transform of the sequence = A134763, (same as a(n) but with interpolated two's).

Links

Crossrefs

Cf. A000108, A000984, A134758, A134759, A134760, A134761, A134763, A134770, A134771.

Programs

  • Magma
    [3*(n+1)*Catalan(n)-2: n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[3*Binomial[2*n,n]-2, {n,0,40}] (* G. C. Greubel, May 28 2024 *)
  • PARI
    a(n) = 3*binomial(2*n, n) - 2; \\ Michel Marcus, Nov 22 2013
  • SageMath
    [3*binomial(2*n,n) -2 for n in range(41)] # G. C. Greubel, May 28 2024

Formula

G.f.: 3/sqrt(1-4*x) - 2/(1-x). - Sergei N. Gladkovskii, Nov 21 2013
From G. C. Greubel, May 28 2024: (Start)
a(n) = 3*(n+1)*A000108(n) - 2.
a(n) = (2*(2*n-1)*a(n-1) + 2*(3*n-2))/n.
E.g.f.: 3*exp(2*x)*BesselI(0, 2*x) - 2*exp(x). (End)

Extensions

More terms from Michel Marcus, Nov 22 2013

A134763 a(n) = (1/2)*( (1+(-1)^n)*A134762(n/2) + 2*(1-(-1)^n) ).

Original entry on oeis.org

1, 2, 4, 2, 16, 2, 58, 2, 208, 2, 754, 2, 2770, 2, 10294, 2, 38608, 2, 145858, 2, 554266, 2, 2116294, 2, 8112466, 2, 31201798, 2, 120349798, 2, 465352558, 2, 1803241168, 2, 7000818658, 2, 27225405898, 2, 106035791398, 2, 413539586458, 2, 1614773623318, 2, 6312296891158, 2
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Comments

Second inverse binomial transform of A134762.
A134762 interpolated with two's.
Former name: A000718^(-2) * A134762. - G. C. Greubel, May 28 2024

Examples

			First few terms of the sequence are: (1, 2, 4, 2, 16, 2, 58, ...), interpolating two's in the sequence A134762: (1, 4, 16, 58, ...).

Links

Crossrefs

Cf. A000718, A000984, A001405, A134758, A134759, A134760, A134761, A134762, A134770, A134771.

Programs

  • Magma
    [3*((n+1) mod 2)*Binomial(n, Floor(n/2)) - 2*(-1)^n : n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[(3/2)*(1+(-1)^n)*Binomial[n,n/2] -2*(-1)^n, {n,0,40}] (* G. C. Greubel, May 28 2024 *)
  • SageMath
    [3*((n+1)%2)*binomial(n, n//2) - 2*(-1)^n for n in range(41)] # G. C. Greubel, May 28 2024

Formula

From G. C. Greubel, May 28 2024: (Start)
a(n) = (1/2)*( (1+(-1)^n)*A134762(n/2) + 2*(1-(-1)^n) ).
a(n) = (3/2)*(1+(-1)^n)*A001405(n) - 2*(-1)^n.
G.f.: 3/sqrt(1-4*x^2) - 2/(1+x).
E.g.f.: 3*BesselI(0, 2*x) - 2*exp(-x). (End)

Extensions

Name change and terms a(14) onward added by G. C. Greubel, May 28 2024

A363660 a(n) = Sum_{d|n} binomial(d+n,n).

Original entry on oeis.org

2, 9, 24, 90, 258, 1043, 3440, 13419, 48850, 187836, 705444, 2725099, 10400614, 40233015, 155133856, 601820876, 2333606238, 9079958260, 35345263820, 137876637843, 538259060526, 2104292500739, 8233430727624, 32248866496625, 126410606580284
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A007503, A116963, A363628.
Cf. A000984, A134758, A343548, A343549, A363661.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + n, n] &]; Array[a, 25] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+n, n));

Formula

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

A383476 Numbers k such that binomial(2k,k) + k is prime.

Original entry on oeis.org

1, 3, 5, 23, 55, 61, 191, 1933, 3049, 8011, 10589, 58687, 100469
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 28 2025

Keywords

Comments

Corresponding primes start 3, 23, 257, 8233430727623, 98527218530093856775578873054487.

Crossrefs

Cf. A134758, A309289.

Programs

  • Magma
    [k: k in [1..1000] | IsPrime(Binomial(2*k,k)+k)];
  • Mathematica
    Select[Range[3500], PrimeQ[Binomial[2*#, #] + #] &] (* Amiram Eldar, Apr 28 2025 *)
  • Python
    from math import comb
from gmpy2 import is_prime
def ok(n): return is_prime(comb(2*n, n) + n)
print([k for k in range(3050) if ok(k)]) # Michael S. Branicky, Apr 28 2025

Extensions

a(7) inserted and a(10)-a(13) from Michael S. Branicky, Apr 29 2025
Showing 1-5 of 5 results.

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