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.

Previous Showing 21-27 of 27 results.

A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 2, 16, 128, 1068, 9142, 79612, 701864, 6244892, 55962920, 504375396, 4567003520, 41513817444, 378596616452, 3462411408136, 31742042431048, 291616814436124, 2684123914512280, 24746511514749280, 228491677484832896, 2112549277665243328
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A072547, A371813.
Cf. A005810, A262977.
Cf. A066381, A262977, A385498, A386701.

Programs

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

Formula

a(n) = [x^n] 1/((1+x) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, -n], [1-4*n], -1). - Stefano Spezia, Apr 07 2024
From Vaclav Kotesovec, Apr 07 2024: (Start)
Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(415*n^3 - 1898*n^2 + 2871*n - 1436)*a(n) = (838715*n^6 - 5099533*n^5 + 12225995*n^4 - 14652035*n^3 + 9157250*n^2 - 2799192*n + 322560)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(415*n^3 - 653*n^2 + 320*n - 48)*a(n-2).
a(n) ~ 2^(8*n + 1/2) / (5 * sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n,k). - Seiichi Manyama, Jul 30 2025
G.f.: g/((-1+2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-4+6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A386960 a(n) = Sum_{k=0..n} 8^k * binomial(2*n,n-k).

Original entry on oeis.org

1, 10, 102, 1036, 10502, 106380, 1077276, 10908096, 110447046, 1118286172, 11322685172, 114642332232, 1160754172316, 11752638152824, 118995469654968, 1204829162684136, 12198895398209862, 123513816397462524, 1250577392936568708, 12662096110945862856, 128203723152486704052
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A032443, A072547, A088218, A100192, A100193.

Programs

  • Magma
    [&+[8^k * Binomial(2*n, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 13 2025
  • Mathematica
    Table[Sum[8^k*Binomial[2*n,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 13 2025 *)
  • PARI
    a(n) = sum(k=0, n, 8^k*binomial(2*n, n-k));

Formula

a(n) = [x^n] 1/((1-9*x) * (1-x)^n).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k-1,n-k).
G.f.: (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).

A109078 Number of symmetric Dyck paths of semilength n and having no hills (i.e., no peaks at level 1).

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 13, 22, 46, 80, 166, 296, 610, 1106, 2269, 4166, 8518, 15792, 32206, 60172, 122464, 230252, 467842, 884236, 1794196, 3406104, 6903352, 13154948, 26635774, 50922986, 103020253, 197519942, 399300166, 767502944, 1550554582
Offset: 0

Views

Author

Emeric Deutsch, Jun 17 2005

Keywords

Comments

Column 0 of A109077.

Examples

			a(4)=4 because we have uudduudd, uudududd, uuududdd and uuuudddd, where u=(1,1), d=(1,-1).

Links

Crossrefs

Cf. A109077.
Bisections are A026641 and A072547.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2*(1-x-x*Sqrt(1-4*x^2)+2*x^2 +Sqrt(1-4*x^2))/(1+Sqrt(1-4*x^2)-2*x)/(1+Sqrt(1-4*x^2)+2*x^2) )); // G. C. Greubel, Apr 29 2019
  • Maple
    g:=2*(1-z-z*sqrt(1-4*z^2)+2*z^2+sqrt(1-4*z^2))/(1+sqrt(1-4*z^2)-2*z)/(1+sqrt(1-4*z^2)+2*z^2): gser:=series(g,z=0,39): 1, seq(coeff(gser,z^n),n=1..36);
  • Mathematica
    CoefficientList[Series[2*(1-x-x*Sqrt[1-4*x^2]+2*x^2+Sqrt[1-4*x^2])/(1+ Sqrt[1-4*x^2]-2*x)/(1+Sqrt[1-4*x^2]+2*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
  • PARI
    my(x='x+O('x^40)); Vec(2*(1-x-x*sqrt(1-4*x^2)+2*x^2 +sqrt(1-4*x^2))/(1+sqrt(1-4*x^2)-2*x)/(1+sqrt(1-4*x^2)+2*x^2)) \\ G. C. Greubel, Mar 16 2017
  • Sage
    (2*(1-x-x*sqrt(1-4*x^2)+2*x^2 +sqrt(1-4*x^2))/(1+sqrt(1-4*x^2)-2*x)/(1+sqrt(1-4*x^2)+2*x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

Formula

G.f.: 2*(1 -z +2*z^2 +(1-z)*q)/((1-2*z+q)*(1+2*z^2+q)), where q = sqrt(1-4*z^2).
a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence 4*(n+1)*a(n) +2*(-n-3)*a(n-1) +2*(-7*n+11)*a(n-2) +(7*n-27)*a(n-3) +2*(-4*n+5)*a(n-4) +4*(n-3)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

A368488 a(n) = Sum_{k=0..n} n^k * binomial(k+n-1,k).

Original entry on oeis.org

1, 2, 17, 334, 10417, 442276, 23690809, 1530206742, 115636017473, 10004657077468, 974950612575601, 105653682110368492, 12602144701834193521, 1640558582759557298696, 231448351542446473323113, 35173958220088874039434726, 5728588740444710703061240065
Offset: 0

Views

Author

Seiichi Manyama, Dec 26 2023

Keywords

Crossrefs

Main diagonal of A368487.
Cf. A000984, A072547.

Programs

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

Formula

a(n) = [x^n] 1/((1-x) * (1-n*x)^n).
a(n) ~ 2^(2*n-1) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Dec 27 2023

A188289 Binomial sum related to rooted trees.

Original entry on oeis.org

0, 2, 3, 14, 45, 167, 609, 2270, 8517, 32207, 122463, 467843, 1794195, 6903353, 26635773, 103020254, 399300165, 1550554583, 6031074183, 23493410759, 91638191235, 357874310213, 1399137067683, 5475504511859, 21447950506395, 84083979575117
Offset: 0

Views

Author

Olivier Gérard, Aug 19 2012

Keywords

Links

Crossrefs

Cf. A000984, A014300, A026641, A178792, A176479, A072547.

Programs

  • GAP
    List([0..30], n-> Binomial(2*n,n) -(-1)^n -Sum([0..n-1], k-> Binomial(2*k,n-1))); # G. C. Greubel, Apr 29 2019
  • Magma
    [n eq 0 select 0 else Binomial(2*n, n) -(-1)^n - (&+[Binomial(2*k, n-1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, Apr 29 2019
  • Mathematica
    Table[Binomial[2n,n]-(-1)^n-Sum[Binomial[2k,n-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Dec 10 2012 *)
  • PARI
    {a(n) = binomial(2*n,n) -(-1)^n -sum(k=0,n-1, binomial(2*k,n-1))}; \\ G. C. Greubel, Apr 29 2019
  • Sage
    [binomial(2*n,n) -(-1)^n -sum(binomial(2*k, n-1) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Apr 29 2019

Formula

a(n) = binomial(2*n,n) - (-1)^n - Sum_{k=0..n-1} binomial(2*k, n-1).
a(n) = Sum_{k=1..n} binomial(n+k,k)*(Sum_{r=n-k..n} (-1)^r*binomial(n-k, r)).
a(n) = (-1)^n*2^(-(1+n))*(1 - 2^(1+n) + (-2)^n*binomial(2+2*n, 1+n) * hypergeometric2F1(1, 2+2*n; 2+n; -1)).
a(n) = Sum_{k=1..n} (-1)^(n+k)*binomial(n+k,k). - Ridouane Oudra, Sep 07 2025

A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 2, 4, 6, 1, 3, 7, 13, 22, 1, 4, 11, 24, 46, 80, 1, 5, 16, 40, 86, 166, 296, 1, 6, 22, 62, 148, 314, 610, 1106, 1, 7, 29, 91, 239, 553, 1163, 2269, 4166, 1, 8, 37, 128, 367, 920, 2083, 4352, 8518, 15792, 1, 9, 46, 174, 541, 1461, 3544, 7896
Offset: 0

Views

Author

Peter Luschny, Nov 14 2012

Keywords

Comments

Notation: If a sequence id is starred then the offset and/or some terms are different. Starred terms indicate the variance.
Row sums: [A026641 ] [1, 1, 4, 13, 46, 166, 610]
--
T(j+2, 2) [A000124*] [1*, 2 , 4, 7, 11, 16, 22]
T(j+3, 3) [A003600*] [1*, 2*, 6, 13, 24, 40, 62]
--
T(j , j) [A072547 ] [1, 0, 2, 6, 22, 80, 296]
T(j+1, j) [A026641 ] [1, 1, 4, 13, 46, 166, 610]
T(j+2, j) [A014300 ] [1, 2, 7, 24, 86, 314, 1163]
T(j+3, j) [A014301*] [1, 3, 11, 40, 148, 553, 2083]
T(j+4, j) [A172025 ] [1, 4, 16, 62, 239, 920, 3544]
T(j+5, j) [A172061 ] [1, 5, 22, 91, 367, 1461, 5776]
T(j+6, j) [A172062 ] [1, 6, 29, 128, 541, 2232, 9076]
T(j+7, j) [A172063 ] [1, 7, 37, 174, 771, 3300, 13820]
--
T(2j ,j) [Central ] [1, 1, 7, 40, 239, 1461, 9076]
T(2j+1,j) [A183160 ] [1, 2, 11, 62, 367, 2232, 13820]
T(2j+2,j) [ ] [1, 3, 16, 91, 541, 3300, 20476]
T(2j+3,j) [A199033*] [1, 4, 22, 128, 771, 4744, 29618]

Examples

			Triangle begins as:
[n]|k->
[0] 1
[1] 1, 0
[2] 1, 1,  2
[3] 1, 2,  4,  6
[4] 1, 3,  7, 13,  22
[5] 1, 4, 11, 24,  46,  80
[6] 1, 5, 16, 40,  86, 166, 296
[7] 1, 6, 22, 62, 148, 314, 610, 1106.

Links

Programs

  • Maple
    A201635 := proc(n,k) option remember; local j;
if n=k then (-1)^n*add(binomial(-n,j), j=0..n)
else add(A201635(n-1,j), j=0..k) fi end:
for n from 0 to 7 do seq(A(n,k), k=0..n) od;
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==n, (-1)^n*Sum[Binomial[-n, j], {j, 0, n}], Sum[T[n-1, j], {j, 0, k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2019 *)
  • PARI
    {T(n,k) = if(k==n, (-1)^n*sum(j=0,n, binomial(-n,j)), sum(j=0,k, T(n-1,j)))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 27 2019
  • Sage
    @CachedFunction
def A201635(n, k):
    if n==k: return (-1)^n*add(binomial(-n, j) for j in (0..n))
    return add(A201635(n-1, j) for j in (0..k))
for n in (0..7) : [A201635(n, k) for k in (0..n)]

A371870 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-k-1,n-2*k).

Original entry on oeis.org

1, 1, 4, 14, 51, 189, 709, 2683, 10220, 39130, 150438, 580328, 2245004, 8705686, 33828704, 131688362, 513445147, 2004688605, 7836832057, 30670416703, 120153739079, 471143251989, 1848978071615, 7261781367389, 28540427527441, 112243216215879, 441693646453729
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2024

Keywords

Crossrefs

Cf. A072547, A114121, A354267.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(2*n-k-1, n-2*k));

Formula

a(n) = [x^n] 1/((1-x-x^2) * (1-x)^(n-1)).
a(n) ~ 4^n / sqrt(Pi*n). - Vaclav Kotesovec, Apr 16 2024
a(n) = A354267(2*n, n). - Peter Luschny, Apr 25 2024
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