A365846
Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^4 ).
Original entry on oeis.org
1, 7, 73, 903, 12281, 177415, 2672377, 41506823, 659972089, 10689904647, 175765581817, 2925998735367, 49219210772473, 835307328307207, 14284937032826873, 245924997499453447, 4258621314671050745, 74128819286282600455, 1296324135131612708857
Offset: 0
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a(n) = sum(k=0, n, binomial(3*n+k+2, k)*binomial(4*(n+1), n-k))/(n+1);
A078990
Triangle arising from (4,2) tennis ball problem, read by rows.
Original entry on oeis.org
1, 1, 2, 3, 1, 4, 10, 16, 22, 1, 6, 21, 52, 105, 158, 211, 1, 8, 36, 116, 301, 644, 1198, 1752, 2306, 1, 10, 55, 216, 678, 1784, 4088, 8144, 14506, 20868, 27230, 1, 12, 78, 360, 1320, 4064, 10896, 25872, 55354, 105704, 183284, 260864, 338444, 1, 14, 105
Offset: 0
Triangle starts:
1;
1, 2, 3;
1, 4, 10, 16, 22;
1, 6, 21, 52, 105, 158, 211;
...
- D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table A.1).
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T(n,k)=if(k<0 || k>2*n,0,if(n<1,k==0,sum(j=0,k,(j+1)*T(n-1,k-j))))
A365848
Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^4 ).
Original entry on oeis.org
1, 9, 122, 1965, 34814, 655290, 12861708, 260312853, 5393696150, 113847928558, 2439377254412, 52919446267698, 1160040801590332, 25655668799151700, 571760925292574640, 12827392114274902629, 289470689505615716070, 6566330844138035042982
Offset: 0
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a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(4*(n+1), n-k))/(n+1);
A181145
G.f.: exp( Sum_{n>=1} [Sum_{k=0..2n} C(2n,k)^2*y^k]*x^n/n ) = Sum_{n>=0,k=0..2n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.
Original entry on oeis.org
1, 1, 4, 1, 1, 12, 27, 12, 1, 1, 24, 134, 236, 134, 24, 1, 1, 40, 410, 1540, 2380, 1540, 410, 40, 1, 1, 60, 975, 6260, 18386, 26216, 18386, 6260, 975, 60, 1, 1, 84, 1981, 19320, 91441, 227052, 306495, 227052, 91441, 19320, 1981, 84, 1, 1, 112, 3612, 49672
Offset: 0
G.f.: A(x,y) = 1 + (1+ 4*y+ y^2)*x + (1 + 12*y+ 27*y^2+ 12*y^3+ y^4)*x^2 + (1+ 24*y+ 134*y^2+ 236*y^3+ 134*y^4+ 24*y^5+ y^6)*x^3 +...
The logarithm of the g.f. begins:
log(A(x,y)) = (1 + 2^2*y + y^2)*x
+ (1 + 4^2*y + 6^2*y^2 + 4^2*y^3 + y^4)*x^2/2
+ (1 + 6^2*y + 15^2*y^2 + 20^2*y^3 + 15^2*y^4 + 6^2*y^5 + y^6)*x^3/3
+ (1 + 8^2*y + 28^2*y^2 + 56^2*y^3 + 70^2*y^4 + 56^2*y^5 + 28^2*y^6 + 8^2*y^7 + y^8)*x^4/4 +...
Triangle begins:
1;
1, 4, 1;
1, 12, 27, 12, 1;
1, 24, 134, 236, 134, 24, 1;
1, 40, 410, 1540, 2380, 1540, 410, 40, 1;
1, 60, 975, 6260, 18386, 26216, 18386, 6260, 975, 60, 1;
1, 84, 1981, 19320, 91441, 227052, 306495, 227052, 91441, 19320, 1981, 84, 1;
1, 112, 3612, 49672, 344260, 1312080, 2883562, 3740572, 2883562, 1312080, 344260, 49672, 3612, 112, 1; ...
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{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,2*m,binomial(2*m,j)^2*y^j)*x^m/m)+O(x^(n+1))),n,x),k,y)}
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