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 1-4 of 4 results.

A134758 a(n) = A000984(n) + n.

Original entry on oeis.org

1, 3, 8, 23, 74, 257, 930, 3439, 12878, 48629, 184766, 705443, 2704168, 10400613, 40116614, 155117535, 601080406, 2333606237, 9075135318, 35345263819, 137846528840, 538257874461, 2104098963742, 8233430727623, 32247603683124, 126410606437777, 495918532948130
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Links

Crossrefs

Cf. A000984, A134759, A134760, A134761, A134762, A134763, A134770, A134771.

Programs

  • Magma
    [n+(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[Binomial[2n,n]+n,{n,0,40}] (* Harvey P. Dale, Dec 10 2011 *)
  • SageMath
    [n+binomial(2*n,n) for n in range(41)] # G. C. Greubel, May 28 2024

Formula

G.f.: ((1-x)^2 + x*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x)). - Harvey P. Dale, Dec 10 2011
From G. C. Greubel, May 28 2024: (Start)
E.g.f.: x*exp(x) + exp(2*x)*BesselI(0, 2*x).
a(n) = (2*(2*n-1)*a(n-1) - (3*n^2 - 6*n + 2))/n. (End)

Extensions

More terms from Harvey P. Dale, Dec 10 2011

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

A141597 Triangle T(n,k) = 2*binomial(n,k)^2 - 1, read by rows, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 17, 17, 1, 1, 31, 71, 31, 1, 1, 49, 199, 199, 49, 1, 1, 71, 449, 799, 449, 71, 1, 1, 97, 881, 2449, 2449, 881, 97, 1, 1, 127, 1567, 6271, 9799, 6271, 1567, 127, 1, 1, 161, 2591, 14111, 31751, 31751, 14111, 2591, 161, 1, 1, 199, 4049, 28799, 88199, 127007, 88199, 28799, 4049, 199, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 21 2008

Keywords

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   7,    1;
  1,  17,   17,     1;
  1,  31,   71,    31,     1;
  1,  49,  199,   199,    49,      1;
  1,  71,  449,   799,   449,     71,     1;
  1,  97,  881,  2449,  2449,    881,    97,     1;
  1, 127, 1567,  6271,  9799,   6271,  1567,   127,    1;
  1, 161, 2591, 14111, 31751,  31751, 14111,  2591,  161,   1;
  1, 199, 4049, 28799, 88199, 127007, 88199, 28799, 4049, 199,  1;

Links

Crossrefs

Cf. A134759 (row sums), A141596.

Programs

  • Magma
    A141597:= func< n,k | 2*Binomial(n,k)^2 - 1 >;
[A141597(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 15 2024
  • Mathematica
    T[n_, m_, k_, l_]:= (1+l)*Binomial[n, m]^k -l;
Table[T[n,m,2,1], {n,0,12}, {m,0,n}]//Flatten
  • SageMath
    def A141597(n,k): return 2*binomial(n,k)^2 -1
flatten([[A141597(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 15 2024

Formula

Sum_{k=0..n} T(n, k) = A134759(n) = 2*binomial(2*n,n) - (n+1) (row sums).
T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k*T(n, k) = ((1+(-1)^n)/2)*(2*(-1)^(n/2)*binomial(n, n/2) - 1) (alternating sign row sums). - G. C. Greubel, Sep 15 2024
Showing 1-4 of 4 results.

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