A079513 -id:A079513 - cp's OEIS frontend

A079514 Second column of triangular array in A079513.

1, 1, 3, 6, 22, 53, 211, 554, 2306, 6362, 27230, 77580, 338444, 986253, 4362627, 12927170, 57788170, 173452334, 781825066, 2370742868, 10757497972, 32892031042, 150073096238, 462030186916, 2117778107732, 6557906929108, 30176799215196, 93909078262808

Offset: 1

N. J. A. Sloane, Jan 22 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table A.2).

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2*(1-Sqrt(1-4*x))/(2+Sqrt(1-4*x)+Sqrt(1+4*x)) )); // G. C. Greubel, Jan 15 2019

Rest[CoefficientList[Series[2*(1-Sqrt[1-4*x])/(2+Sqrt[1-4*x] +Sqrt[1+ 4*x]), {x, 0, 30}], x]] (* G. C. Greubel, Jan 15 2019 *)

my(x='x+O('x^30)); Vec(2*(1-sqrt(1-4*x))/(2+sqrt(1-4*x) +sqrt(1 +4*x))) \\ G. C. Greubel, Jan 15 2019

a=(2*(1-sqrt(1-4*x))/(2+sqrt(1-4*x)+sqrt(1+4*x))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 15 2019

G.f.: 2*(1-sqrt(1-4*x))/(2+sqrt(1-4*x)+sqrt(1+4*x)). - Philippe Deléham, Feb 09 2014

More terms from Philippe Deléham, Feb 09 2014

A066357 Number of ordered (i.e., planar) trees on 2n edges with every subtree at the root having an even number of edges.

Original entry on oeis.org

1, 1, 6, 53, 554, 6362, 77580, 986253, 12927170, 173452334, 2370742868, 32892031042, 462030186916, 6557906929108, 93909078262808, 1355087936016957, 19684187540818866, 287612514032460070, 4224238030616082948, 62329883931236020470, 923519220367120779820

Offset: 0

Louis Shapiro, Feb 01 2002

Row sums of A078990. First column of A079513.
a(n) is the number of walks from (0,0) to (2n,2n) using steps (0,1) and (1,0) which never stray below the line y=x and which avoid the points (m,m) m odd. - Paul Boddington, Mar 14 2003
Series reversion of Sum_{n>0} -a(n)(-x)^n is g.f. of A005900.
a(n) is the number of linear extensions of the one-level grid poset G[(0^n), (1^(n-1)), (1^(n-1))]. The definition of a one-level grid poset can be found in the Pan links. - Ran Pan, Jul 05 2016
These numbers have the same parity as the Catalan numbers C(n), that is, a(n) is even except when n has the form 2^m - 1. This follows immediately from the formula a(n) = C(2*n+1) + 2*C(2*n) - 2^(2*n + 1)*C(n) given below by Callan. We conjecture that a(n) and C(n) have the same 2-adic valuation (checked up to n = 100). - Peter Bala, Aug 02 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Nantel Bergeron, Cesar Ceballos, Vincent Pilaud, Hopf dreams, arXiv:1807.03044 [math.CO], 2018. See p. 17.
A. de Mier and M. Noy, A solution to the tennis ball problem, arXiv:math/0311242 [math.CO], 2003.
J.-G. Luque and J.-Y. Thibon, Noncommutative Symmetric Functions Associated with a Code, Lazard Elimination and Witt Vectors, arXiv:math/0607254 [math.CO], 2006; Discrete Math. Theor. Comput. Sci. 9 (2007), no. 2, 59-72.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (p. 333).
Ran Pan, Problem 1, Project P.
Ran Pan, Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets), Project P.
A. Regev, Enumerating triangulations by parallel diagonals, arXiv:1208.3915 [math.CO], 2012, J. Int. Seq. 15 (2012) #12.8.5

Cf. A078990, A079513, A001448, A005900, A079489, A274644, A274763.

[1] cat [(&+[Binomial(4*n,k)*Binomial(3*n-k-2,n-k-1)/n: k in [0..n]]): n in [1..30]]; // G. C. Greubel, Jan 15 2019

gf := (1-sqrt(1-4*z)-sqrt(1+4*z)+sqrt(1-16*z^2))/(z*(sqrt(1-4*z)-sqrt(1+4*z))):s := series(gf, z, 80): for i from 0 to 50 by 2 do printf(`%d,`,coeff(s,z,i)) od: # James Sellers, Feb 11 2002
a := n -> `if`(n=0,1,binomial(3*n-2,n-1)*hypergeom([1-n,-4*n],[2-3*n], -1)/n): seq(simplify(a(n)),n=0..20); # Peter Luschny, Oct 15 2015

CoefficientList[Series[2/(1 + 4 Sqrt[x]/(Sqrt[1 + 4 Sqrt[x]] - Sqrt[1 - 4 Sqrt[x]])), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

a(n)=local(A); if(n<1,n==0,A=sqrt(1+4*x+O(x^(2*n+2))); A-=subst(A,x,-x); polcoeff(((2*A-8*x)/A^2)^2,2*n))

vector (100, n, n--; if(n<1, 1, sum(k=0, n, binomial(4*n,k)*binomial(3*n-k-2,n-k-1)/n))) \\ Altug Alkan, Oct 07 2015

[1] + [sum(binomial(4*n,k)*binomial(3*n-k-2,n-k-1)/n for k in (0..n)) for n in (1..30)] # G. C. Greubel, Jan 15 2019

For n>0, a(n) = Sum_{r=1..n} C(2*r-1)*a(n-r). Here C(2*r-1) is a Catalan number (A000108). - Paul Boddington, Mar 14 2003
G.f.: 2/(1+4*sqrt(x)/(sqrt(1+4*sqrt(x))-sqrt(1-4*sqrt(x)))).
D-finite with recurrence a(n)*(2*n-1)*(n+1)n = a(n-1)*(32*n^2 - 64*n + 39)*2*n - a(n-2)*(2*n-3)*(4*n-5)*(4*n-7)*16, n>1.
a(0) = 1,a(n) = (1/n)*Sum_{k=0..n} C(4*n,k)*C(3*n-k-2,n-k-1), n>1. - Paul Barry, Apr 09 2007
a(n) = ((2^(4*n))/Gamma(1/2)) * ((6*(2*n+1)*Gamma(2*n+1/2)/Gamma(2*n+3))-2*Gamma(n+1/2)/Gamma(n+2)). - David Dickson (dcmd(AT)unimelb.edu.au), Nov 10 2009
Convolution of A079489 with itself: (1, 6, 53, 554, ...) = (1, 3, 22, 211, ...)*(1, 3, 22, 211, ...).
Proof. Working with Dyck paths, we must show that Dyck paths of size (semilength) 2n, all of whose components (constituent primitive Dyck paths) have even size, are equinumerous with ordered pairs of nonempty Dyck paths of total size 2n in each of which the first component is of odd size and all other components (if any) are of even size. Given a Dyck path P of the former class, use the first return decomposition to write P (uniquely) as the concatenation of U A_1 A_2 ... A_j O E D Q where U denotes upstep, D denotes downstep, A_1,...,A_j are all primitive Dyck paths of even size with j>=0, O is a primitive Dyck path of odd size, E is a Dyck path of even size, and Q is a Dyck path in which all components are of even size. Then P -> (O A_1 A_2 ... A_j, U E D Q) is the desired bijection. QED - David Callan, Apr 11 2012
a(n) = C(2*n+1) + 2*C(2*n) - 2^(2*n+1)*C(n), where C(n) is the Catalan number A000108. This formula can be obtained by manipulating generating functions. The equivalence of this formula and the Barry (Apr 09 2007) sum can be established by the WZ method with a second-order operator. A combinatorial interpretation of the Barry sum would be nice. - David Callan, Apr 10 2012
a(n) ~ (3-2*sqrt(2)) * 2^(4*n) / (n^(3/2) * sqrt(2*Pi)). - Vaclav Kotesovec, Mar 21 2014
exp( Sum_{n >= 1} binomial(4*n,2*n)*x^n/n ) = 1 + 6*x + 53*x^2 + 554*x^3 + ... is an o.g.f. for this sequence omitting the initial term. See A001448. - Peter Bala, Oct 02 2015
a(n) = binomial(3*n-2,n-1)*hypergeom([1-n,-4*n],[2-3*n],-1)/n for n>=1. - Peter Luschny, Oct 15 2015
a(n) = 3*(2*n+1) /(2*n+2) /(4*n+1) *binomial(4*n+2,2*n+1) -4^n /(2*n+1) *binomial(2*n+2,n+1) [Merlini et al F_n formula] - R. J. Mathar, Oct 01 2021

A079515 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 10, 105, 1198, 14506, 183284, 2390121, 31933830, 434920398, 6016012236, 84289034154, 1193717733900, 17060985356980, 245768668712296, 3564709196133737, 52015567131639798, 763050542202081318, 11246882679872658140, 166478073780305341390, 2473696423451621878180

Offset: 0

N. J. A. Sloane, Jan 22 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Table A.3) [Column 1 of this table appears to be incorrect.]

Cf. A079513, A079514, A079516, A079517, A079518, A079519.

c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1-2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r + 1)*c[t]^(r+3); CoefficientList[Series[g[t, 0], {t, 0, 60}], t][[2 ;; ;; 2]] (* G. C. Greubel, Jan 16 2019 *)

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the odd terms in the expansion of g(t,0) = t + 10*t^3 + 105*t^5 + 1198*t^7 + ... - G. C. Greubel, Jan 16 2019

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079516 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 15, 185, 2304, 29482, 386945, 5188169, 70803164, 980545070, 13747777966, 194776025482, 2784380900560, 40113386761524, 581823363803941, 8489505340500521, 124528817146723876, 1835299404114540102, 27163404479642455346, 403573421012802035630

Offset: 0

N. J. A. Sloane, Jan 22 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Table A.3)

Cf. A079513, A079514, A079515, A079517, A079518, A079519.

c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] - (1-2*t)*Sqrt[1+4*t] + Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1)*c[t]^(r+3); Drop[CoefficientList[Series[g[t, 1], {t, 0, 60}], t][[1 ;; ;; 2]], 1] (* G. C. Greubel, Jan 16 2019 *)

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the even terms in the expansion of g(t,1) = 1*t^2 + 15*t^4 + 185*t^6 + 2304*t^8 + ... - G. C. Greubel, Jan 16 2019

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079517 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 21, 301, 4088, 55354, 756059, 10442117, 145803900, 2056351566, 29262470042, 419730456306, 6062949606496, 88127311401876, 1288120149337735, 18922077118169717, 279209456350438708, 4136682188907493702, 61513664658938124486, 917795824360157700870

Offset: 0

N. J. A. Sloane, Jan 22 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Table A.3)

Cf. A079513, A079514, A079515, A079516, A079518.

c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1-2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1)*c[t]^(r+3); Drop[CoefficientList[Series[g[t, 2], {t, 0, 60}], t][[2 ;; ;; 2]], 1] (* G. C. Greubel, Jan 16 2019 *)

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the odd terms in the expansion of g(t,2) = 1*t^3 + 21*t^5 + 301*t^7 + 4088*t^9 + ... - G. C. Greubel, Jan 16 2019

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079518 Coefficients related to tennis ball problem.

Original entry on oeis.org

1, 28, 462, 6832, 97957, 1394180, 19862674, 284156608, 4086496362, 59089988216, 858975619676, 12549322976672, 184195104642157, 2715174884250004, 40181870424263146, 596810833742837536, 8893877150513222014, 132947157383427373320, 1992954280253792526660

Offset: 0

N. J. A. Sloane, Jan 22 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Table A.3)

Cf. A079513, A079514, A079515, A079516, A079517, A079519.

c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1- 2*t)*Sqrt[1+4*t] + Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1)*c[t]^(r+3); Drop[CoefficientList[Series[g[t, 3], {t,0,60}], t][[1;; ;;2]], 2] (* G. C. Greubel, Jan 16 2019 *)

With c(t) = (1 - sqrt(1-4*t))/(2*t), d(t) = (1 -(1+2*t)*sqrt(1-4*t) -(1 - 2*t)*sqrt(1+4*t) + sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r + 1)*c(t)^(r + 3) then the g.f. is given by the even terms in the expansion of g(t,3) = 1*t^4 + 28*t^6 + 462*t^8 + 6832*t^10 + ... - G. C. Greubel, Jan 16 2019

Terms a(5) onward added by G. C. Greubel, Jan 16 2019

A079519 Related to tennis ball problem.

Original entry on oeis.org

12, 284, 5436, 96768, 1664184, 28069444, 467722524, 7730252080, 127023181352, 2078332922360, 33894711502744, 551368536346176, 8950922822411504, 145068948446193428, 2347940754318431196, 37957946888159573968, 613052225104703442120, 9893099103451554441736

Offset: 1

N. J. A. Sloane, Jan 22 2003

nonn

			G.f. = 12*t^2 + 284*t^4 + 5436*t^6 + 96768*t^8 + ... - _G. C. Greubel_, Jan 17 2019

G. C. Greubel, Table of n, a(n) for n = 1..500
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Table A.4)

Cf. A079513, A079514, A079515, A079516, A079517, A079518, A079520.

f[t_]:= Sqrt[1-4*t]; g[t_]:= Sqrt[1+4*t]; S1[t_]:= (1+f[t]-2*f[t]^2)*(1- f[t])^5/(t^3*(f[t]^2-f[t])^2*(2+f[t]+g[t])^2); S3[t_]:= 4*(1-f[t])^2*(1 -g[t])^2*(f[t]^2-(1+2*t)*f[t]-(1-6*t)*g[t]+f[t]*g[t])/(t^3*(2+f[t]+ g[t])^2*(g[t]^2-f[t]-g[t]+f[t]*g[t])^2); W[t_]:= (S1[t]+S1[-t]+S3[t]+ S3[-t])/4; Drop[CoefficientList[Series[W[t], {t, 0, 50}], t][[1 ;; ;; 2]], 1] (* G. C. Greubel, Jan 17 2019 *)

Let f, g, S1 and S3 be given by f(t) = sqrt(1-4*t), g(t) = sqrt(1+4*t), S1(t) = (1+f(t)-2*f(t)^2)*(1- f(t))^5/(t^3*(f(t)^2-f(t))^2*(2+f(t)+g(t))^2), S3(t) = 4*(1-f(t))^2*(1 -g(t))^2*(f(t)^2-(1+2*t)*f(t)-(1-6*t)*g(t)+f(t)*g(t))/(t^3*(2+f(t)+ g(t))^2*(g(t)^2-f(t)-g(t)+ f(t)*g(t))^2). Now let W(t) be given by W(t) = (S1(t) + S1(-t) + S3(t) + S3(-t))/4. The g.f. is the expansion of W(t). - G. C. Greubel, Jan 17 2019

Terms a(5) onward added by G. C. Greubel, Jan 17 2019

A079520 Triangular array related to tennis ball problem, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 10, 0, 1, 5, 15, 31, 0, 1, 6, 21, 52, 105, 0, 1, 7, 28, 80, 185, 343, 0, 1, 8, 36, 116, 301, 644, 1198, 0, 1, 9, 45, 161, 462, 1106, 2304, 4056, 0, 1, 10, 55, 216, 678, 1784, 4088, 8144, 14506, 0, 1, 11, 66, 282, 960, 2744, 6832, 14976, 29482, 50350

Offset: 0

N. J. A. Sloane, Jan 22 2003

Rows have been reversed.

			0.
0, 1.
0, 1, 3.
0, 1, 4, 10.
0, 1, 5, 15, 31.
0, 1, 6, 21, 52, 105. ...

G. C. Greubel, Rows n=0..100 of triangle, flattened
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Fig. A.3)

Leading diagonal gives A079522.

Cf. A079513.

c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1- 2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*t^(r+1) *c[t]^(r+3); Table[SeriesCoefficient[Series[g[t, n-k], {t, 0, n}], n], {n, 0, 12}, {k, 0, n}]//Flatten  (* G. C. Greubel, Jan 17 2019 *)

Let c, d, and g be given by: c(t) = (1-sqrt(1-4*t))/(2*t), d(t) = (1-(1+2*t)*sqrt(1-4*t) -(1- 2*t)*sqrt(1+4*t) +sqrt(1-16*t^2))/(4*t^2), and g(t, r) = d(t)*t^(r+1)*c(t)^(r+3). The rows of the triangle are calculated by the expansion of g(t, n-k) for n>=0, 0 <= k <= n. - G. C. Greubel, Jan 17 2019

Terms a(29) onward added by G. C. Greubel, Jan 17 2019

A079521 Triangular array related to tennis ball problem, read by rows.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 10, 16, 13, 6, 31, 47, 45, 25, 8, 105, 158, 145, 96, 41, 10, 343, 501, 500, 340, 175, 61, 12, 1198, 1752, 1673, 1226, 676, 288, 85, 14, 4056, 5808, 5898, 4326, 2569, 1205, 441, 113, 16, 14506, 20868, 20312, 15608, 9526, 4836, 1987, 640, 145, 18, 50350, 71218, 73000, 55696, 35448, 18800, 8418, 3090, 891, 181, 20

Offset: 0

N. J. A. Sloane, Jan 22 2003

			0.
1,   2.
3,   5,   4.
10,  16,  13,  6.
31,  47,  45,  25, 8.
105, 158, 145, 96, 41, 10. ...

G. C. Greubel, Rows n=0..100 of triangle, flattened
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Fig. A.4).

Leading diagonal gives A079522.

Cf. A079513, A079520, A079521.

c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t]-(1- 2*t)*Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*(t*c[t])^r*(t*c[t]^3 +2*r*c[t]); Table[SeriesCoefficient[Series[g[t, k], {t, 0, n}], n], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Jan 17 2019 *)

Let c, d, and g be given by: c(t) = (1-sqrt(1-4*t))/(2*t), d(t) = (1-(1+ 2*t)*sqrt(1-4*t) -(1-2*t)*sqrt(1+4*t) +sqrt(1-16*t^2))/(4*t^2), and

g(t, r) = d(t)*(t*c(t))^r*(t*c(t)^3 + 2*r*c(t)) then the rows are calculated by the expansion of g(t,k) for n>=0, 0 <= k <= n. - G. C. Greubel, Jan 17 2019

Terms a(28) onward added by G. C. Greubel, Jan 17 2019

A079522 Diagonal of triangular array in A079520.

Original entry on oeis.org

0, 1, 3, 10, 31, 105, 343, 1198, 4056, 14506, 50350, 183284, 647809, 2390121, 8564543, 31933830, 115664164, 434920398, 1588917802, 6016012236, 22134533070, 84289034154, 311957090678, 1193717733900, 4440128821376, 17060985356980, 63732279047612, 245768668712296, 921501110779045

Offset: 0

N. J. A. Sloane, Jan 22 2003

			G.f. = 0 + 1*t + 3*t^2 + 10*t^3 + 31*t^4 + ... - _G. C. Greubel_, Jan 17 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Fig A.3)

Also diagonal of triangular array in A079521.

Cf. A079513, A079520.

F := proc(t) (1-4*t^2-(1+2*t)*sqrt(1-4*t)-(1-2*t)*sqrt(1+4*t)+ sqrt(1-16*t^2))/4/t^3 ; end: d := proc(t) 1+t*F(t) ; end: C := proc(t) (1-sqrt(1-4*t))/2/t ; end: A079521 := proc(h,r) d(t)*t^(r+1)*(C(t))^(r+3) ; expand(%) ; coeftayl(%,t=0,h) ; end: A079522 := proc(n) A079521(n,0) ; end: for n from 0 do printf("%d\n",A079522(n)) ; od: # R. J. Mathar, Sep 20 2009

c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1-2*t)* Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*(t*c[t])^r*(t*c[t]^3 +2*r*c[t]); CoefficientList[Series[g[t, 0], {t, 0, 50}], t] (* G. C. Greubel, Jan 17 2019 *)

Let c, d, and g be given by: c(t) = (1-sqrt(1-4*t))/(2*t), d(t) = (1-(1+ 2*t)*sqrt(1-4*t) -(1-2*t)*sqrt(1+4*t) +sqrt(1-16*t^2))/(4*t^2), and
g(t, r) = d(t)*(t*c(t))^r*(t*c(t)^3 + 2*r*c(t)) then the g.f. is given by the expansion of g(t,0). - G. C. Greubel, Jan 17 2019
a(n) ~ 2^(2*n + 1/2) * (9*sqrt(2) - 10 + (41*sqrt(2) - 58)*(-1)^n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2025

More terms from R. J. Mathar, Sep 20 2009

Terms a(23) onward added by G. C. Greubel, Jan 17 2019

cp's OEIS Frontend

A079514 Second column of triangular array in A079513.

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A066357 Number of ordered (i.e., planar) trees on 2n edges with every subtree at the root having an even number of edges.

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A079515 Coefficients related to tennis ball problem.

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A079516 Coefficients related to tennis ball problem.

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A079517 Coefficients related to tennis ball problem.

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A079518 Coefficients related to tennis ball problem.

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A079519 Related to tennis ball problem.

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A079520 Triangular array related to tennis ball problem, read by rows.