A049235 -id:A049235 - cp's OEIS frontend

A078995 a(n) = Sum_{k=0..n} C(4k,k)C(4*(n-k),n-k).

1, 8, 72, 664, 6184, 57888, 543544, 5113872, 48180456, 454396000, 4288773152, 40503496536, 382701222296, 3617396099936, 34203591636048, 323492394385824, 3060238763412072, 28955508198895584, 274018698082833760, 2593539713410178528, 24550565251665845664

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Y_n for s=4).
Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.7.6.

See A049235 for more information.
Cf. A006256, A079563, A079678, A079679.
Cf. A005810, A147855, A226733, A226761, A385605, A386811.

series(eval(g/(3*g-4), g=RootOf(g = 1+x*g^4,g))^2, x=0, 30); # Mark van Hoeij, May 06 2013

Table[Sum[Binomial[4*k, k]*Binomial[4*(n - k), n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2012 *)

a(n) = sum(k=0, n, binomial(4*k, k)*binomial(4*(n-k), n-k)); \\ Michel Marcus, May 09 2020

a(n) = 2/3*(256/27)^n*(1+c/sqrt(n)+o(n^-1/2)) where c = 2/3*sqrt(2/(3*Pi)) = 0.307105910641187... More generally, a(n, m)=sum(k=0, n, binomial(m*k, k)*binomial(m*(n-k), n-k)) is asymptotic to 1/2*m/(m-1)*(m^m/(m-1)^(m-1))^n. See A000302, A006256 for cases m=2 and 3. - Benoit Cloitre, Jan 26 2003, corrected and extended by Vaclav Kotesovec, Nov 06 2012
243*n*(8*n - 17)*(3*n - 1)*(3*n - 4)*(3*n - 2)*(3*n - 5)*a(n) = 72*(3*n - 5)*(3*n - 4)*(6912*n^4 - 33120*n^3 + 58256*n^2 - 47798*n + 15309)*a(n - 1) - 3072*(2*n - 3)*(6912*n^5 - 55008*n^4 + 175696*n^3 - 282180*n^2 + 227825*n - 73710)*a(n - 2) + 262144*(n - 2)*(4*n - 7)*(2*n - 3)*(2*n - 5)*(4*n - 9)*(8*n - 9)*a(n - 3). - Vladeta Jovovic, Jul 16 2004
Shorter recurrence: 81*n*(3*n-2)*(3*n-1)*(8*n-11)*a(n) = 24*(4608*n^4-14400*n^3+15776*n^2-7346*n+1215)*a(n-1) - 2048*(2*n-3)*(4*n-5)*(4*n-3)*(8*n-3)*a(n-2). - Vaclav Kotesovec, Nov 06 2012
a(n) = Sum_{k=0..n} binomial(4*k+l,k) * binomial(4*(n-k)-l,n-k) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 4^(n-k) * binomial(3*n+k,k). (End)
G.f.: g^2/(3*g-4)^2 where g=ogf(A002293) satisfies g = 1+x*g^4. - Mark van Hoeij, May 06 2013
a(n) = [x^n] 1/((1-4*x) * (1-x)^(3*n+1)). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 15 2025

A031970 Tennis ball problem: Balls 1 and 2 are thrown into a room; you throw one on lawn; then balls 3 and 4 are thrown in and you throw any of the 3 balls onto the lawn; then balls 5 and 6 are thrown in and you throw one of the 4 balls onto the lawn; after n turns, consider all possible collections on lawn and add all the values.

Original entry on oeis.org

0, 3, 23, 131, 664, 3166, 14545, 65187, 287060, 1247690, 5368670, 22917198, 97195968, 410030812, 1722027973, 7204620067, 30044212828, 124932768082, 518215690018, 2144815618522, 8859729437488, 36533517261412, 150410878895818, 618371102344846, 2538971850705064, 10412490129563556

Offset: 0

Louis Shapiro

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ilani Axelrod-Freed, 312-Avoiding Reduced Valid Hook Configurations and Duck Words, arXiv:2010.11834 [math.CO], 2020. See Definition 5.3 p. 14.
Colin L. Mallows and Lou Shapiro, Balls on the Lawn, J. Integer Sequences, Vol. 2, 1999, #5.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.

Cf. A049235, A078516, A079486, A000108.

List([0..30], n-> (2*n^2+5*n+4)*Binomial(2*n+1, n)/(n+2) - 2^(2*n+1)); # G. C. Greubel, Apr 02 2019

[(2*n^2+5*n+4)*Binomial(2*n+1, n)/(n+2) - 2^(2*n+1): n in [0..30]]; // G. C. Greubel, Apr 02 2019

CoefficientList[Series[(1-9*x+20*x^2-(1-7*x+8*x^2)*Sqrt[1-4*x])/(2*x^2*(1 -8*x+16*x^2)), {x,0,30}],x] (* G. C. Greubel, Apr 02 2019 *)

a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1);
/* Joerg Arndt, Dec 04 2012 */

[(2*n^2+5*n+4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1) for n in (0..30)] # G. C. Greubel, Apr 02 2019

a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1). - Colin Mallows.
a(n) = Sum_{i=0..n-1} (4*n-4*i-1)*A028364(n,i), where A028364 is a Catalan triangle. e.g. for n=3 T[3..] = [5,7,9,14] then S(3) = 131 = 11*5 + 7*7 + 3*9. - David Scambler, Apr 27 2009
G.f.: (1-9*x+20*x^2-(1-7*x+8*x^2)*sqrt(1-4*x))/(2*x^2*(1-8*x+16*x^2)). - Vladimir Kruchinin, Apr 02 2019
D-finite with recurrence: (n+2)*a(n) +(-15*n-14)*a(n-1) +2*(40*n-3)*a(n-2) +8*(-22*n+25)*a(n-3) +64*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jan 28 2020

More terms from Joerg Arndt, Dec 04 2012

A075045 Coefficients A_n for the s=3 tennis ball problem.

Original entry on oeis.org

1, 9, 69, 502, 3564, 24960, 173325, 1196748, 8229849, 56427177, 386011116, 2635972920, 17974898872, 122430895956, 833108684637, 5664553564440, 38488954887171, 261369752763963, 1774016418598269, 12035694958994142, 81624256468292016, 553377268856455968

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

T. Amdeberhan, Integrality of a sum.
Roland Bacher, On generating series of complementary plane trees, arXiv:math/0409050 [math.CO], 2004.
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), pp. 307-344 (A_n for s=3).

See A049235 for more information.

Cf. A006256, A036829, A062236, A386617.

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(2,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[2, n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2023, after R. J. Mathar *)

G.f.: seems to be (3*g-1)^(-2)*(1-g)^(-3) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
Conjecture: D-finite with recurrence 8*(2*n+3)*(7*n+1)*(n+1)*a(n) +6*(-252*n^3-477*n^2-220*n-11)*a(n-1) +81*(7*n+8)*(3*n-1)*(3*n+1)*a(n-2)=0. - Jean-François Alcover, Feb 07 2019
a(n) = (3n+2)*(n+1)*binomial(3n+3,n+1)/2/(2n+3) - A049235(n). [Merlini Theorem 2.5 for s=3] - R. J. Mathar, Oct 01 2021
From Seiichi Manyama, Jul 28 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*k+3+l,k) * binomial(3*n-3*k-l,n-k) for every real number l.
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+4,k).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k+3,k). (End)

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

A079486 Number of different solutions to a variant of the 3-ball tennis ball problem.

Original entry on oeis.org

3, 15, 103, 879, 8787, 99061, 1227369, 16409937, 233588249, 3504149013, 54963273921, 895797910129, 15094359120933, 261882874511985, 4662472442136561, 84940003965749601, 1579633610378515989, 29927014639635474589, 576597813697577550447, 11280469732919709557493

Offset: 1

Barry Cipra, Jan 19 2003

You're given labeled balls three at time (labeled 1,2,3 then 4,5,6 etc.), choose two balls from those still on hand (including the three you've just been given) and throw the larger out the window onto the front lawn, the smaller onto the back lawn.

a(n) is the number of 2 X n matrices of positive integers, with the numbers in each row and column increasing, the numbers in column k less than or equal to 3k and no number used more than once.

Kevin Ryde, Table of n, a(n) for n = 1..325
Sean A. Irvine, Java program (github)
Kevin Ryde, PARI/GP Code and Notes

Cf. A000108, A001764, A002293, A049235.

PARI
```
\\ See links.
```

Larger terms through to a(12) computed by Matt Richey.

a(13) onwards from Sean A. Irvine, Aug 17 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

A171074 A115112 with initial term changed from 0 to 1.

Original entry on oeis.org

1, 4, 18, 68, 250, 922, 3430, 12868, 48618, 184754, 705430, 2704154, 10400598, 40116598, 155117518, 601080388, 2333606218, 9075135298, 35345263798, 137846528818, 538257874438, 2104098963718, 8233430727598, 32247603683098

Offset: 1

N. J. A. Sloane, Sep 06 2010

nonn

From a variant of the tennis ball problem (cf. A031970, A049235). On turn n ball 2n-1 is introduced to the room, ball 2n is introduced to the garden, then one of the balls in the room is swapped with one of the balls in the garden. The present sequence gives the number of combinations, while A171075 gives the total on the lawn, A170076 gives the total in the room.

David Scambler, Just for fun, more tennis balls, Posting to the Sequence Fans Mailing List, Aug 25 2010.

Cf. A031970, A049235, A171075, A171076.

cp's OEIS Frontend

A078995 a(n) = Sum_{k=0..n} C(4*k,k)*C(4*(n-k),n-k).

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A075045 Coefficients A_n for the s=3 tennis ball problem.

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

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A079486 Number of different solutions to a variant of the 3-ball tennis ball problem.

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PARI

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

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Maple

Mathematica

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A171074 A115112 with initial term changed from 0 to 1.

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A078995 a(n) = Sum_{k=0..n} C(4k,k)C(4*(n-k),n-k).