A078995 -id:A078995 - cp's OEIS frontend

A226761 G.f.: 1 / (1 + 12xG(x)^2 - 13xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 1, 16, 118, 1004, 8601, 75076, 662796, 5903676, 52949332, 477533356, 4326309406, 39343725716, 358943047438, 3283745710968, 30112624408488, 276715616909148, 2547523969430508, 23491659440021920, 216942761366305144, 2006084011596742384, 18572529488934397689

Offset: 0

Paul D. Hanna, Jun 16 2013

nonn

			G.f.: A(x) = 1 + x + 16*x^2 + 118*x^3 + 1004*x^4 + 8601*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 12*x*G(x)^2 - 13*x*G(x)^3).

Vincenzo Librandi and Joerg Arndt, Table of n, a(n) for n = 0..200

Cf. A002293.

Cf. A005810, A078995, A147855, A226733, A385605, A386811.

Table[Sum[Binomial[2*n+3*k,n-k]*Binomial[2*n-3*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 17 2013 *)

{a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+12*x*G^2-13*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-x*G^2-13*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(2*n+3*k, n-k)*binomial(2*n-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(3*k, n-k)*binomial(4*n-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(4*n+3*k, n-k)*binomial(-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..n} C(3*k, n-k) * C(4*n-3*k, k).
a(n) = Sum_{k=0..n} C(n+3*k, n-k) * C(3*n-3*k, k).
a(n) = Sum_{k=0..n} C(2*n+3*k, n-k) * C(2*n-3*k, k).
a(n) = Sum_{k=0..n} C(3*n+3*k, n-k) * C(n-3*k, k).
a(n) = Sum_{k=0..n} C(4*n+3*k, n-k) * C(-3*k, k).
G.f.: 1 / (1 - x*G(x)^2 - 13*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+5/2)/(7*3^(3*n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 17 2013
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+3*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-3+4*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 7*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A378484 Expansion of (Sum_{k>=0} binomial(4k,k) x^k)^4.

Original entry on oeis.org

1, 16, 208, 2480, 28176, 310336, 3344688, 35472672, 371570320, 3853862080, 39650662720, 405221752112, 4117879215472, 41643345090240, 419362920305952, 4207604570770752, 42079232716865424, 419609034657373120, 4173470598366784960, 41413032430984848832, 410071444666659404352

Offset: 0

Seiichi Manyama, Nov 28 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000302, A378483.

Cf. A005810, A078995.

nmax = 20; CoefficientList[Series[Sum[Binomial[4*k,k] * x^k, {k, 0, nmax}]^4, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 19 2025 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(4*k, k)*x^k)^4)

a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 0} binomial(4*i,i) * binomial(4*j,j) * binomial(4*k,k) * binomial(4*l,l).
G.f.: B(x)^4 where B(x) is the g.f. of A005810.
27*a(n) - 256*a(n-1) = 18*A078995(n) + 8*A005810(n) for n > 0.
a(n) ~ n * 2^(8*n + 2) / 3^(3*n + 2) * (1 + 2^(7/2)/(3^(3/2)*sqrt(Pi*n))). - Vaclav Kotesovec, Jul 19 2025

A078516 Sum of balls on the lawn for the s=4 tennis ball problem.

Original entry on oeis.org

0, 10, 174, 2298, 27258, 305574, 3309444, 35022618, 364559760, 3748221288, 38170570414, 385768464918, 3874673308452, 38718126671076, 385227806897448, 3818752082440794, 37735160423265504, 371852044352248824, 3655440051907792536, 35857177310350860328

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

D. Merlini, R. Sprugnoli, and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (S_n for s=4).

See A049235 for more information.

Cf. A002293, A078995.

a(n) is asymptotic to c*sqrt(n)*(256/27)^n with c=2.8... - Benoit Cloitre, Jan 26 2003

a(n) = ( (4*n^2+11*n+8) * A002293(n+1) - A078995(n+1) ) / 2. - Sean A. Irvine, Jul 02 2025

A078999 Coefficients A_n for the s=4 tennis ball problem.

Original entry on oeis.org

1, 14, 156, 1622, 16347, 161970, 1588176, 15465222, 149866020, 1447117432, 13935821924, 133921143546, 1284811863298, 12309517103724, 117803253946752, 1126336913303526, 10760609522499660, 102733711144434216, 980250448431562864, 9348504508099893272

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.
D. Merlini, R. Sprugnoli, and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (A_n for s=4).

See A049235 for more information.

FussArea := proc(s,n)
    local a,i,j ;
    a := binomial((s+1)*n,n)*n/(s*n+1) ; ;
    add(j *(n-j) *binomial((s+1)*j,j) *binomial((s+1)*(n-j),n-j) /(s*j+1) /(s*(n-j)+1),j=0..n) ;
    a := a+binomial(s+1,2)*% ;
    for j from 0 to n-1 do
        for i from 0 to j do
            i*(j-i) /(s*i+1) /(s*(j-i)+1) /(n-j)
            *binomial((s+1)*i,i) *binomial((s+1)*(j-i),j-i)
            *binomial((s+1)*(n-j)-2,n-1-j) ;
            a := a-%*binomial(s+1,2) ;
        end do:
    end do:
    a ;
end proc:
seq(FussArea(3,n),n=1..30) ; # R. J. Mathar, Mar 31 2023

FussArea[s_, n_] := Module[{a, i, j, pc}, a = Binomial[(s + 1)*n, n]*n/(s*n + 1); pc = Sum[j*(n - j)*Binomial[(s + 1)*j, j]*Binomial[(s + 1)*(n - j), n - j]/(s*j + 1)/(s*(n - j) + 1), {j, 0, n}]; a = a + Binomial[s + 1, 2]*pc; For[j = 0, j <= n - 1 , j++, For[i = 0, i <= j, i++, pc = i*(j - i)/(s*i + 1)/(s*(j - i) + 1)/(n - j)*Binomial[(s + 1)*i, i]* Binomial[(s + 1)*(j - i), j - i]*Binomial[(s + 1)*(n - j) - 2, n - 1 - j]; a = a - pc*Binomial[s + 1, 2]; ]]; a];
Table[FussArea[3, n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2023, after R. J. Mathar *)

Conjecture D-finite with recurrence -729*(3*n+2)*(447758283*n-407746117) *(3*n+4) *(n+1)*a(n) +216*(182049960672*n^4 +605681769096*n^3 -358290749358*n^2 -265170598015*n -38328134998)*a(n-1) +1536 *(30350980224*n^4 -947048676672*n^3 +1377152586736*n^2 -569141632910*n +54868443093)*a(n-2) -131072*(4*n-5) *(351198196*n -151260957) *(4*n-7) *(2*n-3)*a(n-3)=0. - R. J. Mathar, Mar 31 2023

a(n) = ( A078995(n+1) - (5*n+6) * A002293(n+1) ) / 2. - Sean A. Irvine, Jul 02 2025

A268315 Decimal expansion of 256/27.

Original entry on oeis.org

9, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8

Offset: 1

Gheorghe Coserea, Feb 01 2016

			9.481481481481481481481481481481...

Exponential growth rate for A000260, A002293, A005810, A006633, A006634, A052203, A066381, A069271, A078516, A078995, A153396, A196678, A268316.

[9] cat &cat[[4, 8, 1]^^45]; // Vincenzo Librandi, Feb 04 2016

Join[{9}, PadRight[{}, 120, {4, 8, 1}]] (* Vincenzo Librandi, Feb 04 2016 *)

PARI
```
1.0 * 256/27
```

More digits from Jon E. Schoenfield, Mar 15 2018

A358050 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(kj,j) binomial(k*(n-j),n-j).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 16, 4, 0, 1, 8, 39, 64, 5, 0, 1, 10, 72, 258, 256, 6, 0, 1, 12, 115, 664, 1719, 1024, 7, 0, 1, 14, 168, 1360, 6184, 11496, 4096, 8, 0, 1, 16, 231, 2424, 16265, 57888, 77052, 16384, 9, 0, 1, 18, 304, 3934, 35400, 195660, 543544, 517194, 65536, 10, 0

Offset: 0

Seiichi Manyama, Oct 31 2022

			Square array begins:
  1, 1,    1,     1,     1,      1, ...
  0, 2,    4,     6,     8,     10, ...
  0, 3,   16,    39,    72,    115, ...
  0, 4,   64,   258,   664,   1360, ...
  0, 5,  256,  1719,  6184,  16265, ...
  0, 6, 1024, 11496, 57888, 195660, ...

Column k=0-7 give: A000007, A001477(n+1), A000302, A006256, A078995, A079678, A079679, A079563.
Main diagonal gives A358145.
Cf. A358146.

T(n, k) = sum(j=0, n, binomial(k*j, j)*binomial(k*(n-j), n-j));

T(n, k) = sum(j=0, n, (k-1)^(n-j)*binomial(k*n+1, j));

T(n, k) = sum(j=0, n, k^(n-j)*binomial((k-1)*n+j, j));

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

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

A378503 Expansion of (Sum_{k>=0} binomial(4k,k) x^k)^3.

Original entry on oeis.org

1, 12, 132, 1396, 14436, 147120, 1483996, 14854968, 147821604, 1464031120, 14443875984, 142042418004, 1393053544508, 13630170286224, 133092301736232, 1297274743175856, 12624909478998948, 122692158505386960, 1190859983017752880, 11545524234978791952, 111820579340839270416

Offset: 0

Seiichi Manyama, Nov 28 2024

nonn

Cf. A005810, A078995, A337291, A378484.

Cf. A002457, A378483.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(4*k, k)*x^k)^3)

a(n) = Sum_{i+j+k=n, i,j,k >= 0} binomial(4*i,i) * binomial(4*j,j) * binomial(4*k,k).
G.f.: B(x)^3 where B(x) is the g.f. of A005810.
27*a(n) - 256*a(n-1) = 18*A005810(n) - A337291(n) for n > 0.

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 9, 84, 790, 7452, 70401, 665692, 6298236, 59612556, 564393460, 5344664400, 50621130078, 479513718116, 4542730477758, 43039907282664, 407809863233592, 3864303038901996, 36619104142640460, 347027703183853552, 3288802989845088504, 31169274939274755312

Offset: 0

Seiichi Manyama, Jul 28 2025

nonn

Cf. A078995, A308523, A386565, A386611, A386612.

a(n) = sum(k=0, n-1, binomial(4*k-3, k)*binomial(4*n-4*k, n-k-1));

G.f.: (g-1)/(g * (4-3*g)^2) where g=1+x*g^4.
G.f.: g * (1-g)^2/(1-4*g)^2 where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k-3+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n-2,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-2,k).

cp's OEIS Frontend

A226761 G.f.: 1 / (1 + 12*x*G(x)^2 - 13*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A378484 Expansion of (Sum_{k>=0} binomial(4*k,k) * x^k)^4.

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Mathematica

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A078516 Sum of balls on the lawn for the s=4 tennis ball problem.

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A078999 Coefficients A_n for the s=4 tennis ball problem.

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A268315 Decimal expansion of 256/27.

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A358050 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j) * binomial(k*(n-j),n-j).

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A378503 Expansion of (Sum_{k>=0} binomial(4*k,k) * x^k)^3.

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A385251 a(n) = Sum_{k=0..n-1} binomial(4*k-3,k) * binomial(4*n-4*k,n-k-1).

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A226761 G.f.: 1 / (1 + 12xG(x)^2 - 13xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A378484 Expansion of (Sum_{k>=0} binomial(4k,k) x^k)^4.

A358050 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(kj,j) binomial(k*(n-j),n-j).

A378503 Expansion of (Sum_{k>=0} binomial(4k,k) x^k)^3.

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).