A183160
a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k).
Original entry on oeis.org
1, 2, 11, 62, 367, 2232, 13820, 86662, 548591, 3498146, 22436251, 144583496, 935394436, 6071718512, 39523955552, 257913792342, 1686627623151, 11050540084902, 72522925038257, 476669316338542, 3137209052543927, 20672732229560032, 136374124374593072, 900541325129687272
Offset: 0
G.f.: A(x) = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 +...
A(x)^(1/2) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...+ A183161(n)*x^n +...
Given triangle A085478(n,k) = C(n+k,n-k), which begins:
1;
1, 1;
1, 3, 1;
1, 6, 5, 1;
1, 10, 15, 7, 1;
1, 15, 35, 28, 9, 1; ...
ILLUSTRATE formula a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k):
a(2) = 11 = 1*1 + 3*3 + 1*1;
a(3) = 62 = 1*1 + 6*5 + 5*6 + 1*1;
a(4) = 367 = 1*1 + 10*7 + 15*15 + 7*10 + 1*1;
a(5) = 2232 = 1*1 + 15*9 + 35*28 + 28*35 + 9*15 + 1*1; ...
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[(&+[Binomial(n+k, 2*k)*Binomial(2*n-k, k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 22 2021
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Table[Sum[Binomial[n+k,n-k]Binomial[2n-k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)
Table[HypergeometricPFQ[{-n, -n, 1/2 -n, n+1}, {1/2, 1, -2*n}, 1], {n, 0, 25}] (* G. C. Greubel, Feb 22 2021 *)
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{a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(2*n-k,k))}
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{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1-2*x*G^2-3*x^2*G^4), n)} \\ Paul D. Hanna, Nov 03 2012
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{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1+3*x*G-5*x*G^2), n)} \\ Paul D. Hanna, Jun 16 2013
for(n=0, 30, print1(a(n), ", "))
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a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],2)
[simplify(a(n)) for n in range(26)] # Peter Luschny, May 19 2015
A078995
a(n) = Sum_{k=0..n} C(4*k,k)*C(4*(n-k),n-k).
Original entry on oeis.org
1, 8, 72, 664, 6184, 57888, 543544, 5113872, 48180456, 454396000, 4288773152, 40503496536, 382701222296, 3617396099936, 34203591636048, 323492394385824, 3060238763412072, 28955508198895584, 274018698082833760, 2593539713410178528, 24550565251665845664
Offset: 0
- 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.
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series(eval(g/(3*g-4), g=RootOf(g = 1+x*g^4,g))^2, x=0, 30); # Mark van Hoeij, May 06 2013
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Table[Sum[Binomial[4*k, k]*Binomial[4*(n - k), n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2012 *)
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a(n) = sum(k=0, n, binomial(4*k, k)*binomial(4*(n-k), n-k)); \\ Michel Marcus, May 09 2020
A079678
a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=5.
Original entry on oeis.org
1, 10, 115, 1360, 16265, 195660, 2361925, 28577440, 346316645, 4201744870, 51023399190, 620022989200, 7538489480075, 91696845873760, 1115794688036920, 13581508654978560, 165357977228808925, 2013721466517360650, 24527742112263770425, 298805688708113438240, 3640695209795092874290
Offset: 0
- Robert Israel, Table of n, a(n) for n = 0..828
- D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.
- Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.
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seq(add(binomial(5*k,k)*binomial(5*(n-k),n-k),k=0..n), n=0..30); # Robert Israel, Jul 16 2015
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m = 5; Table[Sum[Binomial[m k, k] Binomial[m (n - k), n - k], {k, 0, n}], {n, 0, 17}] (* Michael De Vlieger, Sep 30 2015 *)
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main(size)=my(k,n,m=5); concat(1,vector(size,n, sum(k=0,n, binomial(m*k,k)*binomial(m*(n-k),n-k)))) \\ Anders Hellström, Jul 16 2015
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a(n) = sum(k=0,n,4^(n-k)*binomial(5*n+1,k));
vector(30, n, a(n-1)) \\ Altug Alkan, Sep 30 2015
Original entry on oeis.org
1, 5, 29, 176, 1093, 6885, 43796, 280600, 1807781, 11698223, 75973189, 494889092, 3231947596, 21153123932, 138712176296, 911137377456, 5993760282021, 39481335979779, 260377117268087, 1719026098532296, 11360252318843933, 75141910203168229, 497431016774189912
Offset: 0
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A160906 := proc(n) add( A159841(n,k), k=0..n) ; end:
seq(A160906(n), n=0..20) ;
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Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2017 *)
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a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ Michel Marcus, Oct 31 2017
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a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-1)
[simplify(a(n)) for n in range(21)] # Peter Luschny, May 19 2015
A079679
a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=6.
Original entry on oeis.org
1, 12, 168, 2424, 35400, 520236, 7674144, 113482584, 1681028136, 24932533800, 370144424376, 5499182587416, 81748907485248, 1215834858032820, 18090048027643200, 269246037610828656, 4008495234662771688, 59692297399976544120, 889090275714779739120, 13245013739104555683600
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..851
- D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.
- Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.
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a(n) = sum(k=0,n,5^(n-k)*binomial(6*n+1,k));
vector(30, n, a(n-1)) \\ Altug Alkan, Sep 30 2015
A036829
a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).
Original entry on oeis.org
0, 1, 7, 48, 327, 2221, 15060, 102012, 690519, 4671819, 31596447, 213633696, 1444131108, 9760401756, 65957919496, 445671648228, 3011064814455, 20341769686311, 137412453018933, 928188965638464, 6269358748632207, 42343731580741821
Offset: 0
- M. Petkovsek et al., A=B, Peters, 1996, p. 97.
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a036829 n = sum $ map
(\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
-- Reinhard Zumkeller, May 24 2012
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Table[Sum[Binomial[3k,k]Binomial[3n-3k-2,n-k-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Jan 10 2012 *)
A062236
Sum of the levels of all nodes in all noncrossing trees with n edges.
Original entry on oeis.org
1, 8, 58, 408, 2831, 19496, 133638, 913200, 6226591, 42387168, 288194424, 1957583712, 13286865060, 90126841064, 611029568078, 4140789069408, 28050809681679, 189964288098632, 1286119453570746, 8705397371980728, 58912358137385559, 398607288093924192, 2696583955707785256
Offset: 1
- Harry J. Smith, Table of n, a(n) for n=1..200
- Emeric Deutsch and M. Noy, New statistics on non-crossing trees, in: Formal Power Series and Algebraic Combinatorics (Proceedings of the 12th International Conference, FPSAC'00, Moscow, Russia, 2000), pp. 667-676, Springer, Berlin, 2000.
- Emeric Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87 (see Th. 6). [From _N. J. A. Sloane_, Dec 17 2012]
- P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
- M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
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a := n -> add(2^(n-2-i)*(n-i)*(3*n-3*i-1)*binomial(3*n,i),i=0..n-1)/n;
A062236 := n -> 2^(n-2)*(3*n-1)*hypergeom([-3*n,1-n,-n+4/3], [-n,-n+1/3], -1/2):
seq(simplify(A062236(n)), n = 1..29); # Peter Luschny, Oct 28 2022
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Table[Sum[2^(n-2-k)*(n-k)*(3*n-3*k-1)*Binomial[3*n,k],{k,0,n-1}]/n,{n,1,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
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{ for (n=1, 200, a=sum(i=0, n-1, 2^(n-2-i)*(n-i)*(3*n-3*i-1)*binomial(3*n, i))/n; write("b062236.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 03 2009
A141223
Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108.
Original entry on oeis.org
1, 5, 24, 113, 526, 2430, 11166, 51105, 233190, 1061510, 4822984, 21879786, 99135076, 448707992, 2029215114, 9170247393, 41416383366, 186957126702, 843575853984, 3804927658878, 17156636097156, 77339426905812, 348553445817084, 1570548863858778, 7075531788285276
Offset: 0
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CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2),{x,0,100}],x] (* Emanuele Munarini, Apr 01 2011 *)
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makelist(sum(binomial(n+k,k)*3^(n-k),k,0,n),n,0,12); /* Emanuele Munarini, Apr 01 2011 */
A049235
Sum of balls on the lawn for the s=3 tennis ball problem.
Original entry on oeis.org
0, 6, 75, 708, 5991, 47868, 369315, 2783448, 20631126, 151026498, 1094965524, 7878119760, 56330252412, 400703095284, 2838060684483, 20027058300144, 140874026880204, 988194254587242, 6915098239841331, 48285969880645908, 336521149274459979
Offset: 0
The four sequences T_n, Y_n, A_n, S_n for s=2 are
A000108,
A000302,
A000346,
A031970, for s=3,
A001764,
A006256,
A075045, this sequence, for s=4,
A002293,
A078995,
A078999,
A078516.
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T := (n,s)->binomial(s*n,n)/((s-1)*n+1); Y := (n,s)->add(binomial(s*k,k)*binomial(s*(n-k),n-k),k=0..n); A := (n,s)->Y(n+1,s)/2-(1/2)*((2*s-3)*n+2*s-2)*T(n+1,s); S := (n,s)->(1/2)*(s*n^2+(3*s-1)*n+2*s)*T(n+1,s)-Y(n+1,s)/2;
F := 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3); G := t*(t-1)^2; Ginv := RootOf(G-x,t);
ogf := series(eval(F,t=Ginv), x=0, 20);
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a[n_] := a[n] = Switch[n, 0, 0, 1, 6, 2, 75, 3, 708, 4, 5991, _, -((1/(8*(2*(n-5)^2 + 25*(n-5) + 78)))*(-(531441*(n-5)^2* a[n-5]) + 196830*(n-5)^2*a[n-4] - 24057*(n-5)^2*a[n-3] + 1809*(n-5)^2*a[n-2] - 232*(n-5)^2*a[n-1] - 1594323*(n-5)*a[n-5] + 747954*(n-5)*a[n-4] - 120285*(n-5)*a[n-3] + 16362*(n-5)*a[n-2] - 2798*(n-5)*a[n-1] - 1180980*a[n-5] + 656100*a[n-4] - 131220*a[n-3] + 36825*a[n-2] - 8352*a[n-1]))];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2023, after Robert Israel *)
A385605
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k).
Original entry on oeis.org
1, 7, 58, 502, 4436, 39687, 358024, 3249288, 29624796, 271080124, 2487835678, 22888216006, 211010997716, 1948830506578, 18026768864736, 166976297995452, 1548523206590364, 14376415735219572, 133599985919343400, 1242638966005222648, 11567295503871866536
Offset: 0
Showing 1-10 of 20 results.
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