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 1-10 of 13 results. Next

A119259 Central terms of the triangle in A119258.

Original entry on oeis.org

1, 3, 17, 111, 769, 5503, 40193, 297727, 2228225, 16807935, 127574017, 973168639, 7454392321, 57298911231, 441739706369, 3414246490111, 26447737520129, 205272288591871, 1595964714385409, 12427568655368191, 96905907580960769, 756583504975757311, 5913649000782757889
Offset: 0

Views

Author

Reinhard Zumkeller, May 11 2006

Keywords

Comments

The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 06 2022

References

  • R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Links

Crossrefs

Cf. A005408, A056220, A000225, A000337, A055580, A027608, A119258, A064062, A178792, A014300, A098665.

Programs

  • Haskell
    a119259 n = a119258 (2 * n) n  -- Reinhard Zumkeller, Aug 06 2014
  • Mathematica
    Table[Binomial[2k - 1, k] Hypergeometric2F1[-2k, -k, 1 - 2k, -1], {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
  • Python
    from itertools import count, islice
def A119259_gen(): # generator of terms
    yield from (1,3)
    a, c = 2, 1
    for n in count(1):
        yield (a<>1
A119259_list = list(islice(A119259_gen(),20)) # Chai Wah Wu, Apr 26 2023

Formula

a(n) = A119258(2*n,n).
a(n) = Sum_{k=0..n} C(2*n,k)*C(2*n-k-1,n-k). - Paul Barry, Sep 28 2007
a(n) = Sum_{k=0..n} C(n+k-1,k)*2^k. - Paul Barry, Sep 28 2007
2*a(n) = A064062(n)+A178792(n). - Joseph Abate, Jul 21 2010
G.f.: (4*x^2+3*sqrt(1-8*x)*x-5*x)/(sqrt(1-8*x)*(2*x^2+x-1)-8*x^2-7*x+1). - Vladimir Kruchinin, Aug 19 2013
a(n) = (-1)^n - 2^(n+1)*binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2],2). - Peter Luschny, Jul 25 2014
a(n) = (-1)^n + 2^(n+1)*A014300(n). - Peter Luschny, Jul 25 2014
a(n) = [x^n] ( (1 + x)^2/(1 - x) )^n. Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 13*x^2 + 67*x^3 + ... is essentially the o.g.f. for A064062. - Peter Bala, Oct 01 2015
The o.g.f. is the diagonal of the bivariate rational function 1/(1 - t*(1 + x)^2/(1 - x)) and hence is algebraic by Stanley 1999, Theorem 6.33, p.197. - Peter Bala, Aug 21 2016
n*(3*n-4)*a(n) +(-21*n^2+40*n-12)*a(n-1) -4*(3*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017
From Peter Bala, Mar 23 2020: (Start)
a(p) == 3 ( mod p^3 ) for prime p >= 5. Cf. A002003, A103885 and A156894.
More generally, we conjecture that a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. (End)
G.f.: (8*x)/(sqrt(1-8*x)*(1+4*x)-1+8*x). - Fabian Pereyra, Jul 20 2024
a(n) = 2^(n+1)*binomial(2*n,n) - A178792(n). - Akiva Weinberger, Dec 06 2024
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*n,k). - Seiichi Manyama, Jul 31 2025

A116881 Row sums of triangle A116880 (generalized Catalan C(1,2)).

Original entry on oeis.org

1, 4, 23, 150, 1037, 7408, 54035, 399850, 2990105, 22540260, 170991647, 1303789534, 9983164453, 76711854040, 591236890667, 4568611684306, 35382196437041, 274564234870732, 2134337640202295, 16617270658727878
Offset: 0

Views

Author

Wolfdieter Lang, Mar 24 2006

Keywords

Links

Crossrefs

Cf. A116880, A119259, A178792.

Programs

  • Mathematica
    CoefficientList[Series[(32 x^2 + 12 Sqrt[1 - 8 x] x - 4 x) / (-32 x^3 + Sqrt[1 - 8 x] (8 x^2 + 7 x - 1) - 36 x^2 - 3 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 23 2014 *)
  • Maxima
    a(n):=sum(((k+1)^2*binomial(2*(n+1),n-k)*binomial(n+k+2,n+1))/((n+k+1)*(n+k+2)),k,0,n); /* Vladimir Kruchinin, Nov 23 2014 */

Formula

a(n) = Sum_{m=0..n} A116880(n,m), n>=0.
G.f.: (32*x^2+12*sqrt(1-8*x)*x-4*x)/(-32*x^3+sqrt(1-8*x)*(8*x^2+7*x-1)-36*x^2-3*x+1). - Vladimir Kruchinin, Nov 23 2014
a(n) = sum(k=0..n, ((k+1)^2*binomial(2*(n+1),n-k)*binomial(n+k+2,n+1))/((n+k+1)*(n+k+2))). - Vladimir Kruchinin, Nov 23 2014
a(n) ~ 2^(3*n+3) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 23 2014
Conjecture: n*(3*n-4)*a(n) +(-21*n^2+43*n-10)*a(n-1) -4*(3*n-1)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 22 2016
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] (1+x)^(2*n+1)/(1-x)^n.
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(2*n+1,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(n+k-1,k). (End)

A383326 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k,n-k).

Original entry on oeis.org

1, 7, 71, 799, 9439, 114687, 1419263, 17791487, 225172991, 2870945791, 36819740671, 474470776831, 6138443497471, 79681448443903, 1037278449106943, 13536444411412479, 177030837540093951, 2319618918724403199, 30444928900076666879, 400189735705069486079, 5267487129636270243839
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Crossrefs

Cf. A178792, A383716.
Cf. A370097.

Programs

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

Formula

a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(2*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k).

A383716 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(4*n-k,n-k).

Original entry on oeis.org

1, 9, 127, 2001, 33151, 565249, 9819391, 172826369, 3071424511, 54992986113, 990477877247, 17925526679553, 325710362673151, 5938147061596161, 108571788661555199, 1990032340043366401, 36554697970011340799, 672749920475758460929, 12402180156683794251775
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Crossrefs

Cf. A178792, A383326.
Cf. A370101.

Programs

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

Formula

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(3*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k,k).

A359066 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).

Original entry on oeis.org

1, 1, 5, 7, 31, 49, 209, 351, 1471, 2561, 10625, 18943, 78079, 141569, 580865, 1066495, 4361215, 8085505, 32978945, 61616127, 250806271, 471556097, 1916280833, 3621830655, 14698053631, 27902803969, 113104519169, 215530668031, 872801042431, 1668644405249, 6751535300609
Offset: 1

Views

Author

Bridget Tenner, Dec 15 2022

Keywords

Comments

For n >= 3, this is the number of admissible pinnacle sets in the group S_n^B of signed permutations.
The even-indexed terms appear in A240721 and the odd-indexed terms appear in A178792.

Examples

			For n = 3, the a(3) = 5 admissible pinnacle sets in S_3^B are {}, {-1}, {1}, {2}, {3}.

Links

Crossrefs

Cf. A289871, A359067, A359068, A240721, A178792.

Programs

  • Maple
    a := n -> add(binomial(n, k)*binomial(n-1-k, iquo(n-1, 2) - k), k = 0..iquo(n-1,2)):
# Alternative:
a := n -> binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1-n)/2)], [1-n], -1);
seq(simplify(a(n)), n=3..31); # Peter Luschny, Jan 03 2023
  • Mathematica
    Array[Sum[Binomial[#, k]*Binomial[# - 1 - k, Floor[(# - 1)/2] - k], {k, 0, Floor[(# - 1)/2]}] &, 31] (* Michael De Vlieger, Jan 03 2023 *)
  • PARI
    a(n) = sum(k=0, (n-1)\2, binomial(n,k)*binomial(n-1-k, (n-1)\2 - k)) \\ Andrew Howroyd, Jan 02 2023

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).
a(n) = binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1 -n)/2)], [1 - n], -1). - Peter Luschny, Jan 03 2023

A240721 Expansion of -(4*x + sqrt(1-8*x) - 1)/(sqrt(1-8*x)*(4*x^2+x) + 8*x^2 - x).

Original entry on oeis.org

1, 7, 49, 351, 2561, 18943, 141569, 1066495, 8085505, 61616127, 471556097, 3621830655, 27902803969, 215530668031, 1668644405249, 12944666918911, 100598145875969, 783027553697791, 6103529011806209, 47636654222999551, 372225072921837569, 2911581699143892991
Offset: 0

Views

Author

Vladimir Kruchinin, Apr 11 2014

Keywords

Links

Crossrefs

Cf. A178792.

Programs

  • Maple
    a := n -> binomial(2*n+2,n)*hypergeom([-n, n+2], [n+3],-1);
seq(round(evalf(a(n), 32)), n=0..19); # Peter Luschny, Jul 16 2014
  • Mathematica
    CoefficientList[Series[-(4 x + Sqrt[1 - 8 x] - 1)/(Sqrt[1 - 8 x] (4 x^2 + x) + 8 x^2 - x), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 12 2014 *)
  • Maxima
    a(n) := sum((k+1)*binomial(2*(n+1), n-k)*binomial(n+k+1,n), k, 0, n)/(n+1);
  • Maxima
    a[0]:1$ a[1]:7$ a[2]:49$ a[n] := 8*sum(a[k]*a[n-3-k], k, 0, n-3)+7*sum(a[k]*a[n-2-k], k, 0, n-2)-sum(a[k]*a[n-1-k], k, 0, n-1)+8*a[n-1]$ makelist(a[n], n, 0, 1000); /* Tani Akinari, Jul 16 2014 */
  • PARI
    x='x+O('x^50); Vec(-(4*x+sqrt(1-8*x)-1)/(sqrt(1-8*x)*(4*x^2+x)+8*x^2-x)) \\ G. C. Greubel, Apr 05 2017

Formula

a(n) = (Sum_{k=0..n} (k+1)*binomial(2*(n+1),n-k)*binomial(n+k+1,n))/(n+1).
a(n) = Sum_{k=0..n} binomial(2*(n+1),k)*2^k*(-1)^(n+k) = binomial(2*(n+1),n+1)*(n+1)*Sum_{k=0..n} binomial(n,k)/(n+k+2). - Max Alekseyev, Jun 16 2021
A(x) = (x*B'(x)+B(x))/(x*B(x)+1) where B(x) = (1-4*x-sqrt(1-8*x))/(8*x^2) is the g.f. of A003645.
a(n) ~ 2^(3*n+3)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 12 2014
a(n) = C(2*n+2, n)*2F1([-n, n+2], [n+3], -1), 2F1 is the hypergeometric function. - Peter Luschny, Jul 16 2014
a(n) = 8*Sum_{k=0..n-3} a(k)*a(n-3-k) + 7*Sum_{k=0..n-2} a(k)*a(n-2-k) - Sum_{k=0..n-1} a(k)*a(n-1-k) + 8*a(n-1) for n > 2, a(0)=1, a(1)=7, a(2)=49. - Tani Akinari, Jul 16 2014
D-finite with recurrence -(n+1)*(3*n-2)*a(n) +(21*n^2-5*n-2)*a(n-1) +4*(3*n+1)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
a(n) = 2^(n+1)*binomial(2*n+1,n) - A178792(n). - Akiva Weinberger, Dec 04 2024

A359067 a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k). a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).

Original entry on oeis.org

0, 1, 4, 7, 28, 49, 199, 351, 1436, 2561, 10499, 18943, 77617, 141569, 579149, 1066495, 4354780, 8085505, 32954635, 61616127, 250713893, 471556097, 1915928117, 3621830655, 14696701553, 27902803969, 113099318869, 215530668031, 872780984131, 1668644405249, 6751457741849
Offset: 1

Views

Author

Bridget Tenner, Dec 15 2022

Keywords

Comments

For n >= 3, the number of admissible pinnacle sets in the group S_n^D of even-signed permutations.
The even-indexed terms match the even-indexed terms of A359066. The odd-indexed terms differ from the odd-indexed terms of A359066 by binomial(2*n-1, n).

Examples

			For n = 3, the a(3) = 4 admissible pinnacle sets in S_3^D are {}, {1}, {2}, {3}.

Links

  • Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO] (2023).

Crossrefs

Cf. A289871, A359066, A359068, A178792, A240721.

Programs

  • Maple
    a := n -> if irem(n - 1, 2) = 1 then binomial(n, n/2 - 1)*hypergeom([n/2 + 1, -n/2 + 1], [n/2 + 2], -1) else binomial(n + 1, n/2 + 1/2)*hypergeom([n/2 + 1/2, -n/2 + 1/2], [n/2 + 3/2], -1)/2 - binomial(n - 2, n/2 - 1/2) fi:
seq(simplify(a(n)), n = 3..31); # Peter Luschny, Jan 03 2023

Formula

a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k).
a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).
a(n) = A240721((n-2)/2) if n-1 is odd and otherwise A178792((n-1)/2) - binomial(2*n - 1, n). - Peter Luschny, Jan 03 2023

A386843 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 6, 39, 268, 1905, 13842, 102123, 761880, 5732325, 43417630, 330620895, 2528772132, 19412942809, 149497184298, 1154365194195, 8934458916912, 69291946278861, 538372925816886, 4189702003359687, 32651982699233340, 254800541773725633, 1990683254889381954
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386844, A386845.
Cf. A059304, A178792.
Cf. A116881.

Programs

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

Formula

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

A386898 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(5*n-k,n-k).

Original entry on oeis.org

1, 11, 199, 4031, 85919, 1885311, 42154111, 955020287, 21847988735, 503573013503, 11675986431999, 272033089535999, 6363380561141759, 149354395882487807, 3515589114309115903, 82957940541503045631, 1961823306198598418431, 46482660516543479939071
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Cf. A386812, A386895, A386896, A386897.
Cf. A178792, A383326, A383716.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(5*n-k, n-k));

Formula

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(4*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(4*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n+k,k).
a(n) = binomial(5*n, n)*hypergeom([-1-5*n, -n], [-5*n], -1). - Stefano Spezia, Aug 07 2025

A115197 Convolution of generalized Catalan numbers A064062 (called C(n;2)).

Original entry on oeis.org

1, 2, 7, 32, 169, 974, 5947, 37820, 247885, 1662890, 11362399, 78806936, 553386097, 3926523782, 28108587139, 202764451700, 1472446595221, 10755543924578, 78973277044903, 582558618222416, 4315238786662585
Offset: 0

Views

Author

Wolfdieter Lang, Feb 23 2006

Keywords

Comments

Row sums of triangle A115193, called C(1,2).
The o.g.f. given below follows from the Riordan matrix structure of the triangle A115193. See the o.g.f. for the row polynomials of A115193.

Links

Formula

a(n)= sum(A115193(n,m),m=0..n), n>=0.
G.f.: ((1+2*x*c(2*x))/(1+x))^2 = ((1-2*x) + 6*x*c(2*x))/(1+x)^2, with the o.g.f. c(x) of Catalan numbers A000108.
a(n)= sum(C(2;n-k)*C(2;k),k=0..n), n>=0, with C(2;n):= A064062(n).
a(n)=4*A178792(n)-3*(n+1)*A064062(n+1) [From Joseph Abate, Jun 21 2010]
n*a(n) +(-7*n+13)*a(n-1) +4*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 09 2017
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