Displaying 1-10 of 11 results found.
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
+10
36
1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0
COMMENTS
The signed triangular matrix T(n,k)*(-1)^(n-k) is the inverse matrix of the triangular Catalan convolution matrix A106566(n,k), n=k>=0, with A106566(n,k) = 0 if n<k. - Philippe Deléham, Aug 01 2005 Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n) = Sum_{i=0..k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A111808. For example, s_1(n) = binomial(n,1) = n is the first column of A111808 for n>1, s_2(n) = binomial(n,1) + binomial(n,2) is the second column of A111808 for n>1, etc. Therefore, in cases k=3,4,5,6,7,8, s_k(n) is A005581(n), A005712(n), A000574(n), A005714(n), A005715(n), A005716(n), respectively. Besides, s_k(n+5) = A064054(n). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012 As a triangle, T(n,k) = binomial(k,n-k). - Peter Bala, Nov 27 2015 For all n >= 0, k >= 0, the k-th homology group of the n-torus H_k(T^n) is the free abelian group of rank T(n,k) = binomial(n,k). See the Math Stack Exchange link below. - Jianing Song, Mar 13 2023
LINKS
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
FORMULA
As a number triangle, this is defined by T(n,0) = 0^n, T(0,k) = 0^k, T(n,k) = T(n-1,k-1) + Sum_{j, j>=0} = (-1)^j*T(n-1,k+j)* A000108(j) for n>0 and k>0. - Philippe Deléham, Nov 07 2005 As a triangle read by rows, it is [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 22 2006 As a number triangle, this is defined by T(n, k) = Sum_{i=0..n} (-1)^(n+i)binomial(n, i)binomial(i+k, i-k) and is the Riordan array ( 1, x*(1+x) ). The row sums of this triangle are F(n+1). - Paul Barry, Jun 21 2004 Sum_{k=0..n}x^k*T(n,k) = A000007(n), A000045(n+1), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for n=0,1,2,3,4,5,6,7,8,9,10. - Philippe Deléham, Oct 16 2006 G.f. for the triangular interpretation: -1/(-1+x*y+x^2*y). - R. J. Mathar, Aug 11 2015 For T(0,0) = 0, the triangle below has the o.g.f. G(x,t) = [t*x(1+x)]/[1-t*x(1+x)]. See A109466 for a signed version and inverse, A030528 for reverse and A102426 for a shifted version. - Tom Copeland, Jan 19 2016
EXAMPLE
Array begins
1 0 0 0 0 0 ...
1 1 0 0 0 0 ...
1 2 1 0 0 0 ...
1 3 3 1 0 0 ...
1 4 6 4 1 0 ...
As a triangle, this begins
1
0 1
0 1 1
0 0 2 1
0 0 1 3 1
0 0 0 3 4 1
0 0 0 1 6 5 1
...
Production array is
0 1
0 1 1
0 -1 1 1
0 2 -1 1 1
0 -5 2 -1 1 1
0 14 -5 2 -1 1 1
0 -42 14 -5 2 -1 1 1
0 132 -42 14 -5 2 -1 1 1
0 -429 132 -42 14 -5 2 -1 1 1
MAPLE
seq(seq(binomial(k, n-k), k=0..n), n=0..12); # Peter Luschny, May 31 2014
MATHEMATICA
Table[Binomial[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 28 2015 *)
PROG
(Magma) /* As triangle: */ [[Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 29 2015 (GAP) nmax:=15;; T:=List([0..nmax], n->List([0..nmax], k->Binomial(n, k)));;
b:=List([2..nmax], n->OrderedPartitions(n, 2));;
a:=Flat(List([1..Length(b)], i->List([1..Length(b[i])], j->T[b[i][j][1]][b[i][j][2]]))); # Muniru A Asiru, Jul 17 2018
CROSSREFS
The official entry for Pascal's triangle is A007318. See also A052553 (the same array read by upward antidiagonals). Cf. A030528 (subtriangle for 1<=k<=n).
Coefficient of x^4 in expansion of (1+x+x^2)^n.
(Formerly M4129)
+10
30
1, 6, 19, 45, 90, 161, 266, 414, 615, 880, 1221, 1651, 2184, 2835, 3620, 4556, 5661, 6954, 8455, 10185, 12166, 14421, 16974, 19850, 23075, 26676, 30681, 35119, 40020, 45415, 51336, 57816, 64889, 72590, 80955, 90021, 99826, 110409, 121810, 134070
COMMENTS
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-3) is the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007 Antidiagonal sums of the convolution array A213781. [Clark Kimberling, Jun 22 2012]
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: (x^2)*(1+x-x^2)/(1-x)^5.
a(n) = binomial(n+2,n-2) + binomial(n+1,n-2) - binomial(n,n-2). - Zerinvary Lajos, May 16 2006 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). Vincenzo Librandi, Jun 16 2012 a(n) = GegenbauerC(N, -n, -1/2) where N = 4 if 4<n else 2*n-4. - Peter Luschny, May 10 2016 E.g.f.: exp(x)*x^2*(12 + 12*x + x^2)/24. - Stefano Spezia, Jul 09 2023
MAPLE
seq(binomial(n+2, n-2) + binomial(n+1, n-2) - binomial(n, n-2), n=2..50); # Zerinvary Lajos, May 16 2006 A005712 := n -> GegenbauerC(`if`(4<n, 4, 2*n-4), -n, -1/2):
MATHEMATICA
CoefficientList[Series[(1+x-x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 16 2012 *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 6, 19, 45, 90}, 40] (* Harvey P. Dale, Apr 30 2015 *)
PROG
(Magma) I:=[1, 6, 19, 45, 90]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; Vincenzo Librandi, Jun 16 2012
CROSSREFS
a(n)= A027907(n, 4), n >= 2 (fifth column of trinomial coefficients).
Left half of trinomial triangle ( A027907), triangle read by rows.
+10
30
1, 1, 1, 1, 2, 3, 1, 3, 6, 7, 1, 4, 10, 16, 19, 1, 5, 15, 30, 45, 51, 1, 6, 21, 50, 90, 126, 141, 1, 7, 28, 77, 161, 266, 357, 393, 1, 8, 36, 112, 266, 504, 784, 1016, 1107, 1, 9, 45, 156, 414, 882, 1554, 2304, 2907, 3139, 1, 10, 55, 210, 615, 1452, 2850, 4740, 6765, 8350
COMMENTS
Consider a doubly infinite chessboard with squares labeled (n,k), ranks or rows n in Z, files or columns k in Z (Z denotes ...,-2,-1,0,1,2,... ); number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A026300, A114929, A114972. Triangle of numbers C^(2)(n-1,k), n>=1, of combinations with repetitions from elements {1,2,...,n} over k, such that every element i, i=1,...,n, appears in a k-combination either 0 or 1 or 2 times (cf. also A213742- A213745). - Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012
REFERENCES
Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
FORMULA
(1 + x + x^2)^n = Sum(T(n,k)*x^k: 0<=k<=n) + Sum(T(n,k)*x^(2*n-k): 0<=k<n);
T(n, k) = A027907(n, k) = Sum_{i=0,..,(k/2)} binomial(n, n-k+2*i) * binomial(n-k+2*i, i), 0<=k<=n.
MAPLE
T := (n, k) -> simplify(GegenbauerC(k, -n, -1/2)):
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, May 09 2016
MATHEMATICA
Table[GegenbauerC[k, -n, -1/2], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)
CROSSREFS
T(n, 0) = 0;
T(n, 1) = n for n>1;
Coefficient of x^5 in expansion of (1 + x + x^2)^n.
(Formerly M3011 N1219)
+10
12
3, 16, 51, 126, 266, 504, 882, 1452, 2277, 3432, 5005, 7098, 9828, 13328, 17748, 23256, 30039, 38304, 48279, 60214, 74382, 91080, 110630, 133380, 159705, 190008, 224721, 264306, 309256, 360096, 417384, 481712, 553707, 634032, 723387, 822510
COMMENTS
If Y is a 3-subset of an n-set X then, for n>=7, a(n-4) is the number of 5-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: x^3*(3-2*x)/(1-x)^6.
a(n) = 3*binomial(n+2,5) - 2*binomial(n+1,5).
a(n) = binomial(n+1, 4)*(n+12)/5 = 3*b(n-3)-2*b(n-4), with b(n)=binomial(n+5, 5); cf. A000389. a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Vincenzo Librandi, Jun 10 2012 a(n) = GegenbauerC(N, -n, -1/2) where N = 5 if 5<n else 2*n-5. - Peter Luschny, May 10 2016 E.g.f.: exp(x)*x^3*(60 + 20*x + x^2)/120. - Stefano Spezia, Jul 09 2023
MAPLE
seq(3*binomial(n+2, 5)-2*binomial(n+1, 5), n=3..100); # Robert Israel, Aug 04 2015 A000574 := n -> GegenbauerC(`if`(5<n, 5, 2*n-5), -n, -1/2):
MATHEMATICA
CoefficientList[Series[(3-2*x)/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 10 2012 *)
PROG
(Magma) [3*Binomial(n+2, 5)-2*Binomial(n+1, 5): n in [3..50]]; // Vincenzo Librandi, Jun 10 2012 (PARI) x='x+O('x^50); Vec(x^3*(3-2*x)/(1-x)^6) \\ G. C. Greubel, Nov 22 2017
CROSSREFS
Column m=5 of (1, 3) Pascal triangle A095660.
Coefficient of x^6 in expansion of (1+x+x^2)^n.
(Formerly M4704)
+10
9
1, 10, 45, 141, 357, 784, 1554, 2850, 4917, 8074, 12727, 19383, 28665, 41328, 58276, 80580, 109497, 146490, 193249, 251713, 324093, 412896, 520950, 651430, 807885, 994266, 1214955, 1474795, 1779121, 2133792, 2545224, 3020424, 3567025, 4193322, 4908309, 5721717
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = binomial(n, 3)*(n^3+18*n^2+17*n-120) /120.
G.f.: (x^3)*(1+3*x-4*x^2+x^3)/(1-x)^7. (Numerator polynomial is N3(6, x) from A063420). a(n) = A027907(n, 6), n >= 3 (seventh column of trinomial coefficients). a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7). Vincenzo Librandi, Jun 16 2012 a(n) = GegenbauerC(N, -n, -1/2) where N = 6 if 6<n else 2*n-6. - Peter Luschny, May 10 2016 E.g.f.: exp(x)*x^3*(120 + 180*x + 30*x^2 + x^3)/720. - Stefano Spezia, Mar 28 2023
MAPLE
A005714 := n -> GegenbauerC(`if`(6<n, 6, 2*n-6), -n, -1/2):
MATHEMATICA
a[n_] := Coefficient[(1 + x + x^2)^n, x, 6]; Table[a[n], {n, 3, 35}]
CoefficientList[Series[(1+3*x-4*x^2+x^3)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 16 2012 *)
PROG
(Magma) I:=[1, 10, 45, 141, 357, 784, 1554]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Jun 16 2012 (Magma) /* By definition: */ P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[7]: n in [3..35] ]; // Bruno Berselli, Jun 17 2012
Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).
+10
9
1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 19, 10, 1, 1, 15, 45, 45, 15, 1, 1, 21, 90, 141, 90, 21, 1, 1, 28, 161, 357, 357, 161, 28, 1, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55
COMMENTS
Subtriangle (for 1<=k<=n) of triangle defined by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2006
LINKS
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 6.
FORMULA
T(n, k) = Sum_{j=0..k-1} C(n-1, 2k-j-2)*C(2k-j-2, j).
G.f.: A(x, y) = (1 - x*(1+y))/(1 - 2*x*(1+y) + x^2*(1+y+y^2)) (offset=0). - Paul D. Hanna, Feb 26 2005 G.f.: 1/(1-x-xy-x^2y/(1-x-xy)).
E.g.f.: exp((1+y)x)*cosh(sqrt(y)*x).
T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2*(k-j)). (End)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,1) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<1 or if k>n. - Philippe Deléham, Mar 27 2014
EXAMPLE
Triangle begins:
1;
1,1;
1,3,1;
1,6,6,1;
1,10,19,10,1;
...
Triangle (0, 1, 0, 1, 0, 0, 0...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 1, 6, 6, 1;
0, 1, 10, 19, 10, 1;
0, 1, 15, 45, 45, 15, 1;
0, 1, 21, 90, 141, 90, 21, 1;
MATHEMATICA
t[n_, k_] := Sum[ Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Oct 11 2011, after Paul Barry *)
PROG
(PARI) T(n, k)=if(n<k || k<1, 0, polcoeff((1+x+x^2)^(n-1)+O(x^(2*k)), 2*k-2)) \\ Paul D. Hanna
T(n,k)=Number of nXk 0..2 arrays with rows and columns in lexicographic nondecreasing order but with exactly two mistakes.
+10
7
0, 0, 0, 1, 20, 1, 15, 264, 264, 15, 90, 2550, 9354, 2550, 90, 357, 22267, 201539, 201539, 22267, 357, 1107, 166762, 3576730, 11454780, 3576730, 166762, 1107, 2907, 1046418, 58069125, 514122657, 514122657, 58069125, 1046418, 2907, 6765, 5586207
COMMENTS
Table starts
.....0.........0.............1................15....................90
.....0........20...........264..............2550.................22267
.....1.......264..........9354............201539...............3576730
....15......2550........201539..........11454780.............514122657
....90.....22267.......3576730.........514122657...........62922179364
...357....166762......58069125.......20086951472.........6584300364020
..1107...1046418.....859516239......724313811311.......615691843257769
..2907...5586207...11336482734....24378309172117.....53477639726024161
..6765..25997719..132278417831...757386980723842...4387410446730955493
.14355.107862842.1373129978107.21490393664858691.339567886171232998387
FORMULA
Empirical for column k:
k=1: [polynomial of degree 8]
k=2: [polynomial of degree 26]
k=3: [polynomial of degree 80]
EXAMPLE
Some solutions for n=3 k=4
..1..0..1..2. .0..0..0..0. .0..1..2..0. .0..1..1..2. .1..2..1..2
..0..0..0..2. .1..1..0..1. .0..0..0..1. .0..1..0..2. .0..2..1..0
..2..2..2..1. .0..2..1..1. .2..2..0..1. .2..0..0..2. .0..2..1..1
T(n,k)=Number of nXk 0..2 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly two mistakes.
+10
7
0, 0, 0, 1, 13, 1, 15, 285, 285, 15, 90, 3354, 10824, 3354, 90, 357, 27521, 234484, 234484, 27521, 357, 1107, 175881, 3739008, 10776210, 3739008, 175881, 1107, 2907, 932205, 48592635, 387551595, 387551595, 48592635, 932205, 2907, 6765, 4266912
COMMENTS
Table starts
....0.......0..........1............15................90..................357
....0......13........285..........3354.............27521...............175881
....1.....285......10824........234484...........3739008.............48592635
...15....3354.....234484......10776210.........387551595..........11719632199
...90...27521....3739008.....387551595.......33967584488........2593097277036
..357..175881...48592635...11719632199.....2593097277036......521528860552802
.1107..932205..541463431..309971214338...175644502146694....94782697883923436
.2907.4266912.5325263364.7350438329498.10734760074025367.15624069731088285787
FORMULA
Empirical for column k:
k=1: [polynomial of degree 8]
k=2: [polynomial of degree 24]
k=3: [polynomial of degree 70]
EXAMPLE
Some solutions for n=3 k=4
..0..1..2..1. .0..2..1..0. .0..1..2..2. .1..0..0..1. .0..2..0..2
..1..0..0..0. .2..2..2..1. .1..0..2..2. .0..2..1..0. .2..0..0..0
..2..1..0..0. .2..0..1..2. .1..2..1..0. .2..2..2..2. .2..2..2..1
Coefficient of x^7 in expansion of (1+x+x^2)^n.
(Formerly M3632)
+10
6
4, 30, 126, 393, 1016, 2304, 4740, 9042, 16236, 27742, 45474, 71955, 110448, 165104, 241128, 344964, 484500, 669294, 910822, 1222749, 1621224, 2125200, 2756780, 3541590, 4509180, 5693454, 7133130, 8872231, 10960608, 13454496
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = binomial(n, 4)*(n^3+27*n^2+116*n-120)/210, n >= 4.
G.f.: (x^4)*(x-2)*(x^2-2)/(1-x)^8. (Numerator polynomial is N3(7, x) from A063420). a(n) = A027907(n, 7), n >= 4 (eighth column of trinomial coefficients). a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8). Vincenzo Librandi, Jun 16 2012 a(n) = GegenbauerC(N, -n, -1/2) where N = 7 if 7<n else 2*n-7. - Peter Luschny, May 10 2016
MAPLE
A005715 := n -> GegenbauerC(`if`(7<n, 7, 2*n-7), -n, -1/2):
MATHEMATICA
CoefficientList[Series[(x-2)*(x^2-2)/(1-x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 16 2012 *)
PROG
(Magma) I:=[4, 30, 126, 393, 1016, 2304, 4740, 9042]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5)-28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Jun 16 2012 (Magma) /* By definition: */ P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[8]: n in [4..33] ]; // Bruno Berselli, Jun 17 2012
Tenth column of trinomial coefficients.
+10
3
5, 50, 266, 1016, 3139, 8350, 19855, 43252, 87802, 168168, 306735, 536640, 905658, 1481108, 2355962, 3656360, 5550755, 8260934, 12075184, 17363896, 24597925, 34370050, 47419905, 64662780, 87222720, 116470380, 154066125, 202008896, 262691396, 338962184
LINKS
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
a(n) = binomial(n+5, 5)*(n^4+66*n^3+1307*n^2+8706*n+15120) /(9!/5!).
G.f.: (1+x-x^2)*(5-5*x+x^2)/(1-x)^10, numerator polynomial is N3(9, x)= 5+0*x-9*x^2+6*x^3-x^4 from array A063420. a(n) = GegenbauerC(N, -n, -1/2) where N = 9 if 9<n else 2*n-9. - Peter Luschny, May 10 2016
MAPLE
A064054 := n -> GegenbauerC(`if`(9<n, 9, 2*n-9), -n, -1/2):
MATHEMATICA
Table[GegenbauerC[9, -n, -1/2], {n, 5, 50}] (* G. C. Greubel, Feb 28 2017 *)
PROG
(PARI) for(n=0, 25, print1(binomial(n+5, 5)*(n^4 + 66*n^3 + 1307*n^2 + 8706*n + 15120) /(9!/5!), ", ")) \\ G. C. Greubel, Feb 28 2017
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