A005260 -id:A005260 - cp's OEIS frontend

A216795 a(n) = sum_{k=0..n} binomial(n,k)^4 * 3^k.

1, 4, 58, 1000, 19426, 412744, 9195796, 212836432, 5062716850, 123033947464, 3041489363188, 76243484446672, 1933564156575364, 49518970223489680, 1278877982692134568, 33269141292429734560, 870987510534775369810, 22930499187530338390600, 606700679139764282611540

Offset: 0

Vaclav Kotesovec, Sep 16 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012

Cf. A005260, A216696.

Table[Sum[Binomial[n, k]^4*3^k, {k, 0, n}], {n, 0, 25}]

Recurrence: -(x-1)^4*(n+1)^3*(n+2)*(16*(4*x^2 + 17*x + 4)*n^4 + 184*(4*x^2 + 17*x + 4)*n^3 + (3137*x^2 + 13351*x + 3137)*n^2 + (5867*x^2 + 25051*x + 5867)*n + 4061*x^2 + 17438*x + 4061)*a(n) + (n+2)*(64*(4*x^5 + 141*x^4 + 655*x^3 + 655*x^2 + 141*x + 4)*n^7 + 1024*(4*x^5 + 141*x^4 + 655*x^3 + 655*x^2 + 141*x + 4)*n^6 + 4*(6857*x^5 + 242368*x^4 + 1126775*x^3 + 1126775*x^2 + 242368*x + 6857)*n^5 + 8*(12439*x^5 + 442336*x^4 + 2059985*x^3 + 2059985*x^2 + 442336*x + 12439)*n^4 + (211031*x^5 + 7579744*x^4 + 35400065*x^3 + 35400065*x^2 + 7579744*x + 211031)*n^3 + (261344*x^5 + 9524206*x^4 + 44667470*x^3 + 44667470*x^2 + 9524206*x + 261344)*n^2 + (174888*x^5 + 6498997*x^4 + 30655175*x^3 + 30655175*x^2 + 6498997*x + 174888)*n + 15*(3251*x^5 + 123835*x^4 + 588594*x^3 + 588594*x^2 + 123835*x + 3251))*a(n+1) -(32*(12*x^4-197*x^3-1030*x^2-197*x + 12)*n^8 + 624*(12*x^4-197*x^3-1030*x^2-197*x + 12)*n^7 + 10*(6295*x^4-103673*x^3-542004*x^2-103673*x + 6295)*n^6 + 4*(74418*x^4-1233343*x^3-6448430*x^2-1233343*x + 74418)*n^5 + (864893*x^4-14467663*x^3-75685660*x^2-14467663*x + 864893)*n^4 + 20*(78938*x^4-1336491*x^3-7002327*x^2-1336491*x + 78938)*n^3 + (1764932*x^4-30321697*x^3-159367410*x^2-30321697*x + 1764932)*n^2 + (1102551*x^4-19262286*x^3-101826010*x^2-19262286*x + 1102551)*n + 10*(29405*x^4-523232*x^3-2793306*x^2-523232*x + 29405))*a(n+2) + (n+3)*(64*(4*x^3 + 21*x^2 + 21*x + 4)*n^7 + 1152*(4*x^3 + 21*x^2 + 21*x + 4)*n^6 + 12*(2899*x^3 + 15226*x^2 + 15226*x + 2899)*n^5 + 4*(35609*x^3 + 187226*x^2 + 187226*x + 35609)*n^4 + (340693*x^3 + 1795162*x^2 + 1795162*x + 340693)*n^3 + (474743*x^3 + 2511132*x^2 + 2511132*x + 474743)*n^2 + (355831*x^3 + 1894439*x^2 + 1894439*x + 355831)*n + 10*(11039*x^3 + 59401*x^2 + 59401*x + 11039))*a(n+3) -(n+3)*(n+4)^3*(16*(4*x^2 + 17*x + 4)*n^4 + 120*(4*x^2 + 17*x + 4)*n^3 + (1313*x^2 + 5599*x + 1313)*n^2 + 15*(103*x^2 + 443*x + 103)*n + 659*x^2 + 2882*x + 659)*a(n+4) = 0, this is case x=3.
a(n) ~ (1+3^(1/4))^3/(4*sqrt(2)*3^(3/8)*Pi^(3/2)) * (1+3^(1/4))^(4*n)/n^(3/2). - Vaclav Kotesovec, Sep 19 2012
Generally, Sum_{k=0..n} binomial(n,k)^p*x^k is asymptotic a(n) ~ (1+x^(1/p))^(p*n+p-1)/sqrt((2*pi*n)^(p-1)*p*x^(1-1/p)). This is case p=4, x=3. - Vaclav Kotesovec, Sep 19 2012

Minor edits by Vaclav Kotesovec, Mar 31 2014

A328808 Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

Original entry on oeis.org

1, 3, 23, 225, 2583, 32133, 422069, 5757699, 80790775, 1158593589, 16905540753, 250185539079, 3746205581589, 56652844671855, 864032059578879, 13274539401672345, 205252378269637815, 3191578469685269925, 49876569284504593505, 782943268394316187815

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Column k=4 of A328807.

Cf. A005260, A328725, A328735.

Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^4, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = polcoef(polcoef(polcoef((1+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

{a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)*a(n-1) + (n-1)*(22*n^2 - 44*n + 13)*a(n-2) - 44*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 51*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ sqrt(2) * 17^(n + 3/2) / (64 * Pi^(3/2) * n^(3/2)). (End)

A337332 a(n) = Sum_{k=0..n}C(n,k)C(n+k,k)C(2k,k)C(2n-2k,n-k)(-8)^(n-k).

Original entry on oeis.org

1, -12, 228, -3504, 44580, -298032, 1407504, -275772096, 21324125988, -966349948080, 32198201397648, -831808446595776, 16275197594916624, -210881419152530112, 1110165241205298240, -28746364298042321664, 4877709692143697517348, -323151109677783574203312, 13976671241536620108719376

Offset: 0

Zhi-Wei Sun, Aug 23 2020

sign

(-1)^n*a(n) > 0, and Sum_{k=0..n} C(n,k)*C(n+k,k)*C(2k,k)*C(2n-2k,n-k)*(-1)^(n-k) = Sum_{k=0..n}C(n,k)^4.
Conjecture 1: Sum_{k>=0}(4k+1) a(k)/(-48)^k = sqrt(72+42*sqrt(3))/Pi.
Conjecture 2: For each n > 0, the number (Sum_{k=0..n-1} (-1)^k*(4k+1)*48^(n-1-k)*a(k))/n is a positive integer.
Conjecture 3: For any prime p > 3, the square of (Sum_{k=0..p-1} (4k+1)a(k)/(-48)^k)/p is congruent to 14*(3/p)-(p/3)-12 modulo p, where (a/p) is the Legendre symbol.
Conjecture 4: Let p > 3 be a prime, and let S(p) = Sum_{k=0..p-1} a(k)/(-48)^k. If p == 1 (mod 4) and p = x^2 + 4y^2 with x and y integers, then S(p) == 4x^2-2p (mod p^2). If p == 3 (mod 4), then S(p) == 0 (mod p^2).

			a(1) = C(1,0)*C(1,0)*C(0,0)*C(2,1)*(-8) + C(1,1)*C(2,1)*C(2,1)*C(0,0) = -16 + 4 = -12.

Zhi-Wei Sun, Table of n, a(n) for n = 0..100
Zhi-Wei Sun, An explicit solution to the congruence x^2 == 14*(3/p)-(p/3)-12 (mod p)?, Question 369963 at MathOverflow, August 23, 2020.
Zhi-Wei Sun, New series for powers of Pi and related congruences, Electron. Res. Arch. 28(2020), no. 3, 1273-1342.
Zhi-Wei Sun, Some new series for 1/Pi motivated by congruences, arXiv:2009.04379 [math.NT], 2020.

Cf. A000796, A000984, A005260, A336981, A336982, A337247.

a[n_]:=Sum[Binomial[n,k]Binomial[n+k,k]Binomial[2k,k]Binomial[2(n-k),n-k](-8)^(n-k),{k,0,n}];
Table[a[n],{n,0,18}]

a(n) = (-8)^n*binomial(2*n, n)*hypergeom([1/2, -n, -n, n + 1], [1, 1, 1/2 - n], 1/8). - Peter Luschny, Aug 24 2020

A357603 a(n) is the number of different pairs of shortest paths in an n X n lattice going between opposite corners in opposite directions and not meeting at their middle point.

Original entry on oeis.org

0, 2, 18, 236, 3090, 42252, 589932, 8383608, 120720402, 1756863020, 25789460268, 381298472568, 5671808350572, 84807208655288, 1273785187835640, 19207311526394736, 290631247129611282, 4411188317020786668, 67137528193253129484, 1024357917198436543800

Offset: 0

Janaka Rodrigo, Oct 05 2022

Equivalently, a(n) is the number of different ways to interchange the positions of two men standing at opposite corners of an n X n lattice without meeting each other.
The two men start to move simultaneously at the same constant speed; one always moves to the right or upward, the other always moves to the left or downward.
All terms are even.

			Let the lattice points of a lattice of size 2 X 2 be labeled 1,2,3,4,5,6,7,8,9, and let men A and B start at points 1 and 9, respectively.
                          man B
                          starts
             7---8---9 <-- here
             |   |   |
   man A     4---5---6
   starts    |   |   |
    here --> 1---2---3
.
The lattice paths available for A are 14789, 14589, 14569, 12589, 12569, 12369 and those available for B are 98741, 98541, 98521, 96541, 96521, 96321.
A002894(2) = 36 is the number of different ways to exchange positions, that is, 6 X 6 or (C(4,2))^2.
The different ways they can meet halfway on their paths are as follows:
If A selects 14789, B must select 98741. If A selects one of 14589, 14569, 12589, 12569, B must select one of 98541, 98521, 96541, 96521. If A selects 12369, B must select 96321.
Therefore the total number of choices available is 1 + 4*4 + 1 = 18 and this is given by A005260(2) = (C(2,0))^4 + (C(2,1))^4 + (C(2,2))^4 = 18.
Therefore the total number of such different pairs is a(2) = 36 - 18 = 18.

Cf. A005260, A002894.

a(n) = A002894(n) - A005260(n).

A007804 Related to the asymptotic expansion of Sum_{k = 0..n} C(n,k)^4.

Original entry on oeis.org

1, 30, 1730, 152340, 18177750, 2742927876, 500848449300, 107365679361000, 26431620161451750, 7348772237141884500, 2277376143931016207100, 778374526612873263759000, 290867891728117751744917500

Offset: 0

Bruno Haible

nonn

Cf. A005260.

A334237 a(n) = 2Sum_{k=0..n-1} binomial(n,k)^2binomial(n,k+1)^2.

Original entry on oeis.org

2, 16, 198, 2368, 30100, 392544, 5248782, 71501056, 989177508, 13859716000, 196282985756, 2805235913088, 40408113882344, 586055349387200, 8551024115349150, 125431745952519168, 1848653992986172324, 27362153523832614432, 406546456064695351020

Offset: 1

Bradley Klee, Apr 19 2020

a(n) is also the number of simultaneous walks between two walkers on an n X n grid, subject to a "social distancing" constraint. The rules are the same as in A005260, but the counting criterion is changed so that the walkers cannot meet. Instead, they must be separated by closest-approach distance of sqrt(2) after n steps. Each term a(n) is a hypergeometric single sum, so Zeilberger's algorithm applies, and a(n) must also satisfy a p-recurrence.

B. Klee and É. Angelini, "Social Distancing and A005260", [math-fun] mailing list, Apr. 19, 2020.

Oskar Schlemmer, Das Triadisches Ballett: Gelbe Marsch, Bavaria Atelier, 1970.
D. Zeilberger, The Method of Creative Telescoping, Journal of Symbolic Computation, 11.3 (1991), 195-204.

Cf. A005260.

RecurrenceTable[{Dot[{(n-1)^2*(n+1)^3*(5*n^2-10*n+4),
-2*n^2*(2*n-1)*(15*n^4-30*n^3+7*n^2+8*n-8),
-4*(n-1)^2*n*(4*n-5)*(4*n-3)*(5*n^2-1)},
a[n-#]&/@Range[0,2]] == 0, a[0] == 0, a[1] == 2},
a, {n, 0, 100}]

a(n) = 2*sum(k=0, n-1, binomial(n,k)^2*binomial(n,k+1)^2); \\ Michel Marcus, Apr 19 2020

D-finite with recurrence (n-1)^2*(n+1)^3*(5*n^2-10*n+4)*a(n) - 2*n^2*(2*n-1)*(15*n^4-30*n^3+7*n^2+8*n-8)*a(n-1) - 4*(n-1)^2*n*(4*n-5)*(4*n-3)*(5*n^2-1)*a(n-2) = 0.

a(n) ~ 2^(4*n + 3/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Apr 20 2020

cp's OEIS Frontend

A216795 a(n) = sum_{k=0..n} binomial(n,k)^4 * 3^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A328808 Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A337332 a(n) = Sum_{k=0..n}C(n,k)*C(n+k,k)*C(2k,k)*C(2n-2k,n-k)*(-8)^(n-k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A357603 a(n) is the number of different pairs of shortest paths in an n X n lattice going between opposite corners in opposite directions and not meeting at their middle point.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A007804 Related to the asymptotic expansion of Sum_{k = 0..n} C(n,k)^4.

Views

Author

Keywords

Crossrefs

A334237 a(n) = 2*Sum_{k=0..n-1} binomial(n,k)^2*binomial(n,k+1)^2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328808 Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

A337332 a(n) = Sum_{k=0..n}C(n,k)C(n+k,k)C(2k,k)C(2n-2k,n-k)(-8)^(n-k).

A334237 a(n) = 2Sum_{k=0..n-1} binomial(n,k)^2binomial(n,k+1)^2.