A051286 -id:A051286 - cp's OEIS frontend

A377153 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

1, 6, 27, 140, 651, 2772, 11354, 44640, 169371, 624742, 2248575, 7922124, 27397937, 93214632, 312559200, 1034507696, 3384194616, 10954244952, 35118346760, 111602517096, 351819819414, 1100912299156, 3421515852834, 10566654790176, 32441857824859, 99060134392422

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Cf. A051286, A182884, A377145, A377148, A377152, A377158, A377159.

Cf. A001874, A089627.

a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=5, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..2} A089627(5,k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(11/2).

A377158 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 7, 35, 196, 994, 4578, 20118, 84540, 341397, 1335103, 5078227, 18852428, 68519920, 244413820, 857393700, 2963013816, 10102413972, 34025396580, 113329367816, 373642488044, 1220412680410, 3951964394642, 12695738508950, 40484919514284, 128216539026261

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Cf. A051286, A182884, A377145, A377148, A377152, A377153, A377159.

Cf. A089627.

a(n) = sum(k=0, n, binomial(k+6, 6)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=6, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..3} A089627(6,k) * (1-x-x^2)^(6-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(13/2).

A377159 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 8, 44, 264, 1446, 7152, 33516, 149688, 640233, 2642992, 10582220, 41249000, 157050660, 585621960, 2143442400, 7715164176, 27353809188, 95660348904, 330377130644, 1127996393656, 3810881349814, 12750188169312, 42276102419916, 139008143200536, 453526927536969

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Cf. A051286, A182884, A377145, A377148, A377152, A377153, A377158.

Cf. A089627.

a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=7, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..3} A089627(7,k) * (1-x-x^2)^(7-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(15/2).

A181547 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^5.

Original entry on oeis.org

1, 1, 2, 33, 245, 1268, 10903, 108801, 876184, 7319995, 70550669, 663827604, 6051592703, 57695451167, 563736086740, 5452227384417, 53094611797387, 525962074892014, 5232943624317191, 52145361057635835, 523458523860890906

Offset: 0

Paul D. Hanna, Oct 29 2010

nonn

Conjecture: Given F(n,L) = Sum_{k=0..[n/2]} C(n-k,k)^L, then lim_{n->oo} F(n+1,L)/F(n,L) = (Fibonacci(L)*sqrt(5) + Lucas(L))/2 for L>=0 where Fibonacci(n) = A000045(n) and Lucas(n) = A000032(n).
For this sequence (L=5): lim_{n->oo} a(n+1)/a(n) = (5*sqrt(5)+11)/2 = 11.090...
Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - u)*(1 - v) - (x*y*z*u*v)^2). - Ilya Gutkovskiy, Apr 23 2025

			G.f. A(x) = 1 + x + 2*x^2 + 33*x^3 + 245*x^4 + 1268*x^5 + 10903*x^6 +...
The terms begin:
a(0) = a(1) = 1^5;
a(2) = 1^5 + 1^5 = 2;
a(3) = 1^5 + 2^5 = 33;
a(4) = 1^5 + 3^5 + 1^5 = 245;
a(5) = 1^5 + 4^5 + 3^5 = 1268;
a(6) = 1^5 + 5^5 + 6^5 + 1^5 = 10903;
a(7) = 1^5 + 6^5 + 10^5 + 4^5 = 108801; ...

Seiichi Manyama, Table of n, a(n) for n = 0..963

Cf. variants: A181545, A181546, A051286.

Cf. A000032, A000045.

PARI
```
{a(n)=sum(k=0,n\2,binomial(n-k,k)^5)}
```

A188648 Binomial sums a(n) = Sum_{k=0..n} (binomial(2n-k,k))^2.

Original entry on oeis.org

1, 2, 11, 63, 376, 2317, 14545, 92512, 594169, 3844787, 25027296, 163701327, 1075049011, 7083830648, 46812088751, 310118453573, 2058919125662, 13695571200353, 91254952276859, 608960974528058, 4069232436916151

Offset: 0

Emanuele Munarini, Apr 07 2011

Central coefficients of A172991.

Bisection of A051286 (Whitney number of level n of the lattice of the ideals of the fence of order 2n). - Paul D. Hanna, Apr 07 2011

Seiichi Manyama, Table of n, a(n) for n = 0..1198 (terms 0..121 from Vincenzo Librandi)

Sum_{k=0..n} (binomial(2n-k,k))^b: A122367(n) = A001519(n+1) (b=1), this sequence (b=2).

Cf. A054142, A172991, A051286.

Table[Sum[Binomial[2n-k,k]^2,{k,0,n}],{n,0,20}]
Table[DifferenceRoot[Function[{y, m}, {4 (-m + n)^2 (-1 - 2 m + 2 n)^2 y[m] + (-5 m^2 - 18 m^3 - 17 m^4 + 12 m n + 56 m^2 n + 68 m^3 n - 8 n^2 - 56 m n^2 - 100 m^2 n^2 + 16 n^3 + 64 m n^3 - 16 n^4) y[1 + m] + (1 + m)^2 (-m + 2 n)^2 y[2 + m] == 0, y[0] == 0, y[1] == 1}]][n + 1], {n, 0, 20}] (* Benedict W. J. Irwin, Nov 03 2016 *)

makelist(sum(binomial(2*n-k,k)^2,k,0,n),n,0,20);

{a(n) = sum(k=0, n, binomial(2*n-k, k)^2)} \\ Seiichi Manyama, Jan 13 2019

G.f.: 1/2*(1/sqrt(1-2*sqrt(x)-x-2*x*sqrt(x)+x^2) + 1/sqrt(1+2*sqrt(x)-x+2*x*sqrt(x)+x^2)).
Recurrence: (n-2)*n*(2*n - 1)*(48*n^2 - 192*n + 169)*a(n) = (576*n^5 - 4032*n^4 + 10212*n^3 - 11414*n^2 + 5457*n - 849)*a(n-1) + 5*(2*n - 3)*(48*n^4 - 288*n^3 + 565*n^2 - 399*n + 64)*a(n-2) + (576*n^5 - 4608*n^4 + 13668*n^3 - 18286*n^2 + 10521*n - 1896)*a(n-3) - (n-3)*(n-1)*(2*n - 5)*(48*n^2 - 96*n + 25)*a(n-4). - Vaclav Kotesovec, Mar 02 2014
a(n) ~ phi^(4*n + 2) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2014, simplified Jan 13 2019
Conjecture: a(n) = hypergeom([-n,-n,n+1,n+1], [1/2,1/2,1], 1/16). - Velin Yanev, Oct 31 2019
a(n) = A051286(2*n). - Mark van Hoeij, Sep 05 2022

A051291 Whitney number of level n of the lattice of the ideals of the fence of order 2 n + 1.

Original entry on oeis.org

1, 2, 3, 7, 17, 40, 97, 238, 587, 1458, 3640, 9124, 22951, 57904, 146461, 371281, 943045, 2399460, 6114555, 15603339, 39866932, 101976512, 261117378, 669239402, 1716737267, 4407306170, 11323050897, 29110603423, 74888578067

Offset: 0

Emanuele Munarini

nonn

This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008

			a(2) = 3 because the ideals of size 2 of the fence F(5) = { x1 < x2 > x3 < x4 > x5 } are x1x2, x1x3, x2x3.

E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A051292.

G.f.: function = (1+2*t^2-t^3-(1-t)*sqrt(1-2*t-t^2-2*t^3+t^4))/(2*t*sqrt(1-2*t-t^2-2*t^3+t^4))

A182878 Triangle read by rows: T(n,k) is the number of lattice paths L_n of weight n having length k (0 <= k <= n). These are paths that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 4, 1, 0, 0, 1, 9, 1, 0, 0, 0, 9, 16, 1, 0, 0, 0, 1, 36, 25, 1, 0, 0, 0, 0, 16, 100, 36, 1, 0, 0, 0, 0, 1, 100, 225, 49, 1, 0, 0, 0, 0, 0, 25, 400, 441, 64, 1, 0, 0, 0, 0, 0, 1, 225, 1225, 784, 81, 1, 0, 0, 0, 0, 0, 0, 36, 1225, 3136, 1296, 100, 1, 0, 0, 0, 0, 0, 0, 1, 441, 4900, 7056, 2025, 121, 1

Offset: 0

Emeric Deutsch, Dec 10 2010

The weight of a path is the sum of the weights of its steps.
Sum of entries in row n is A051286(n).
Sum_{k=0..n} k*T(n,k) = A182879(n).

			Denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are hhh, hH, Hh, ud, and du, having lengths 3, 2, 2, 2, and 2, respectively.
Triangle starts:
  1;
  0,  1;
  0,  1,  1;
  0,  0,  4,  1;
  0,  0,  1,  9,  1;
  0,  0,  0,  9, 16,  1;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A182879.

T:=(n,k)->binomial(k,n-k)^2: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T(n,k) = binomial(n,n-k)^2.

G.f. = G(t,z) = ((1-t*z)^2 - 2*t*z^2 - 2*t^2*z^3 + t^2*z^4)^(-1/2).

Keyword tabl added by Michel Marcus, Apr 09 2013

A110198 Antidiagonal sums of number triangle A110197.

Original entry on oeis.org

1, 2, 4, 9, 20, 46, 109, 262, 638, 1569, 3886, 9680, 24225, 60856, 153368, 387573, 981742, 2491934, 6336721, 16139616, 41166912, 105139773, 268841100, 688157430, 1763206441, 4521749642, 11605580290, 29809644693, 76621733444, 197074591420, 507193044993

Offset: 0

Paul Barry, Jul 15 2005

Partial sums of A051286.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A051286, A110197, A201631.

CoefficientList[Series[1/((1-x)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

G.f.: 1/((1-x)*sqrt((1+x+x^2)*(1-3x+x^2))); a(n) = sum{k=0..floor(n/2), sum{i=0..n-2k, binomial(i+k, k)^2}}.
a(n) = sum{i=0..2n, A202411(i)}. - Peter Luschny, Jan 16 2012
Conjecture: n*a(n) +(-3*n+1)*a(n-1) +n*a(n-2) +(-n+2)*a(n-3) +(3*n-5)*a(n-4) +(-n+2)*a(n-5)=0. - R. J. Mathar, Nov 15 2012
a(n) ~ sqrt(100+45*sqrt(5)) * ((sqrt(5)+3)/2)^n / (10*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 08 2014
Equivalently, a(n) ~ phi^(2*n + 3) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

A180717 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2x^k]^2 x^n.

Original entry on oeis.org

1, 1, 3, 10, 37, 140, 544, 2181, 8873, 36647, 152950, 644313, 2734648, 11681428, 50173541, 216532005, 938383331, 4081653710, 17811999929, 77957939080, 342099306436, 1504801777973, 6633574235109, 29300516237855

Offset: 0

Paul D. Hanna, Sep 29 2010

nonn

Compare g.f. to a g.f. of the Whitney numbers (A051286):

. Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k] * x^n.

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 140*x^5 + 544*x^6 +...
equals the sum of the series:
A(x) = 1 + (1 + x)^2*x + (1 + 4*x + x^2)^2*x^2
+ (1 + 9*x + 9*x^2 + x^3)^2*x^3
+ (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^2*x^4
+ (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^2*x^5
+ (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^2*x^6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..900

Cf. A180718.

{a(n)=polcoeff(sum(m=0,n,sum(k=0,m,binomial(m,k)^2*x^k)^2*x^m)+x*O(x^n),n)}

a(n) ~ c * d^n / (Pi*n), where d = 1/3*(5 + (187/2 - (9*sqrt(93))/2)^(1/3) + (1/2*(187 + 9*sqrt(93)))^(1/3)) = 4.61347026758155538... is the root of the equation 1 - 2*d + 5*d^2 - d^3 = 0, c = 1/192*(80 + (382976 - 18432*sqrt(93))^(1/3) + 8*2^(2/3)*(187 + 9*sqrt(93))^(1/3)) = 1.15336756689... is the root of the equation 64*c^3 - 80*c^2 + 8*c - 1 = 0. - Vaclav Kotesovec, Jul 31 2014

A182896 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,-1)-returns to the horizontal axis. The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 2, 4, 1, 8, 3, 17, 9, 37, 25, 1, 82, 66, 5, 185, 171, 20, 423, 437, 70, 1, 978, 1107, 225, 7, 2283, 2790, 686, 35, 5373, 7009, 2015, 147, 1, 12735, 17574, 5760, 553, 9, 30372, 44019, 16135, 1932, 54, 72832, 110210, 44500, 6398, 264, 1, 175502, 275925, 121247, 20350, 1134, 11

Offset: 0

Emeric Deutsch, Dec 12 2010

			T(3,1)=1. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; exactly one of them, namely ud, has one (1,-1)-return to the horizontal axis.
Triangle starts:
   1;
   1;
   2;
   4,  1;
   8,  3;
  17,  9;
  37, 25, 1;
  82, 66, 5;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1750
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A004148, A182897.

eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-t*z^3*c-z^3*c): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do: #yields sequence in triangular form

T(n)={[Vecrev(p) | p<-Vec(1/(1-x-x^2 - (1+y)*(1-x-x^2 - sqrt(1+x^4-2*x^3-x^2-2*x+O(x*x^n)))/2))]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Nov 05 2019

G.f.: G(t,z) = 1/(1-z-z^2-(1+t)z^3*c), where c satisfies c = 1 + zc + z^2*c + z^3*c^2.
Sum of entries in row n is A051286(n).
T(n,0) = A004148(n+1) (the secondary structure numbers).
Sum_{k=0..n} k*T(n,k) = A182897(n).

Data corrected by Andrew Howroyd, Nov 05 2019

cp's OEIS Frontend

A377153 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

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A377158 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(k,n-k)^2.

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A377159 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(k,n-k)^2.

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A181547 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)^5.

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A188648 Binomial sums a(n) = Sum_{k=0..n} (binomial(2n-k,k))^2.

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Maxima

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A051291 Whitney number of level n of the lattice of the ideals of the fence of order 2 n + 1.

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A110198 Antidiagonal sums of number triangle A110197.

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A180717 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2x^k]^2 x^n.