A005716 -id:A005716 - cp's OEIS frontend

A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

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

Offset: 0

N. J. A. Sloane, Jan 19 2003

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 nPhilippe Deléham, Aug 01 2005
As a number triangle: unsigned version of A109466. - Philippe Deléham, Oct 26 2008
A063967*A130595 as infinite lower triangular matrices. - Philippe Deléham, Dec 11 2008
Modulo 2, this sequence becomes A106344. - Philippe Deléham, Dec 18 2008
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

			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
  ... (Cf. A000108)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Tom Copeland, Addendum to Elliptic Lie Triad
Math Stack Exchange, Homology of the n-torus using the Künneth Formula
Lili Mu and Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

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).
Cf. A109466, A102426.

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

/* As triangle: */ [[Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 29 2015

seq(seq(binomial(k,n-k),k=0..n),n=0..12); # Peter Luschny, May 31 2014

Table[Binomial[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 28 2015 *)

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
T(n,k) = A109466(n,k)*(-1)^(n-k). - Philippe Deléham, Dec 11 2008
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

A111808 Left half of trinomial triangle (A027907), triangle read by rows.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 17 2005

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

Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Trinomial Triangle
Eric Weisstein's World of Mathematics, Trinomial Coefficient

Row sums give A027914; central terms give A027908;
T(n, 0) = 0;
T(n, 1) = n for n>1;
T(n, 2) = A000217(n) for n>1;
T(n, 3) = A005581(n) for n>2;
T(n, 4) = A005712(n) for n>3;
T(n, 5) = A000574(n) for n>4;
T(n, 6) = A005714(n) for n>5;
T(n, 7) = A005715(n) for n>6;
T(n, 8) = A005716(n) for n>7;
T(n, 9) = A064054(n-5) for n>8;
T(n, n-5) = A098470(n) for n>4;
T(n, n-4) = A014533(n-3) for n>3;
T(n, n-3) = A014532(n-2) for n>2;
T(n, n-2) = A014531(n-1) for n>1;
T(n, n-1) = A005717(n) for n>0;
T(n, n) = central terms of A027907 = A002426(n).

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

Table[GegenbauerC[k, -n, -1/2], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

(1 + x + x^2)^n = Sum(T(n,k)*x^k: 0<=k<=n) + Sum(T(n,k)*x^(2*n-k): 0<=k

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.

T(n, k) = GegenbauerC(k, -n, -1/2). - Peter Luschny, May 09 2016

Corrected and edited by Johannes W. Meijer, Oct 05 2010

A000574 Coefficient of x^5 in expansion of (1 + x + x^2)^n.

Original entry on oeis.org

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

Offset: 3

N. J. A. Sloane

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

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).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A005581, A005712, A005714-A005716, A111808.
Column m=5 of (1, 3) Pascal triangle A095660.
Cf. A000217, A055998.

[3*Binomial(n+2,5)-2*Binomial(n+1,5): n in [3..50]]; // Vincenzo Librandi, Jun 10 2012

A000574:=-(-3+2*z)/(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation
seq(3*binomial(n+2,5)-2*binomial(n+1,5),n=3..100); # Robert Israel, Aug 04 2015
A000574 := n -> GegenbauerC(`if`(5A000574(n)), n=3..20); # Peter Luschny, May 10 2016

CoefficientList[Series[(3-2*x)/(1-x)^6,{x,0,40}],x] (* Vincenzo Librandi, Jun 10 2012 *)

x='x+O('x^50); Vec(x^3*(3-2*x)/(1-x)^6) \\ G. C. Greubel, Nov 22 2017

G.f.: x^3*(3-2*x)/(1-x)^6.
a(n) = 3*binomial(n+2,5) - 2*binomial(n+1,5).
a(n) = A111808(n,5) for n>4. - Reinhard Zumkeller, Aug 17 2005
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) = 3*binomial(n, 3) + 4*binomial(n, 4) + binomial(n, 5). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 5 if 5Peter Luschny, May 10 2016
a(n) = Sum_{i=1..n-1} A000217(i)*A055998(n-1-i). - Bruno Berselli, Mar 05 2018
E.g.f.: exp(x)*x^3*(60 + 20*x + x^2)/120. - Stefano Spezia, Jul 09 2023

More terms from Vladeta Jovovic, Oct 02 2000

A005714 Coefficient of x^6 in expansion of (1+x+x^2)^n.

Original entry on oeis.org

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

Offset: 3

N. J. A. Sloane

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).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
R. K. Guy, Letter to N. J. A. Sloane, 1987
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000574, A005581, A005712, A005715-A005716, A111808.

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

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[7]: n in [3..35] ]; // Bruno Berselli, Jun 17 2012

A005714:=-(1+3*z-4*z**2+z**3)/(z-1)**7; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005714 := n -> GegenbauerC(`if`(6A005714(n)), n=3..20); # Peter Luschny, May 10 2016

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 *)

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) = A111808(n,6) for n>5. - Reinhard Zumkeller, Aug 17 2005
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) = binomial(n,3) + 6*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 6 if 6Peter 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

More terms from Vladeta Jovovic, Oct 02 2000

A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

Original entry on oeis.org

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

Offset: 1

Colin Mallows, Aug 23 2000

Forms the even-indexed trinomial coefficients (A027907). Matrix inverse is A104027. - Paul D. Hanna, Feb 26 2005

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

			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;
  ... - _Philippe Deléham_, Mar 27 2014

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 6.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 7.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.

Columns are A000217, A005712, A005714, A005716.

Cf. A027907, A104027, A117569, A124302.

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 *)

PARI
```
T(n,k)=if(nPaul D. Hanna
```

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
Sum_{k, 1<=k<=n}T(n,k)=A124302(n). Sum_{k, 1<=k<=n}(-1)^(n-k)*T(n,k)=A117569(n). - Philippe Deléham, Oct 29 2006
From Paul Barry, Sep 28 2010: (Start)
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

More terms from James Sellers, Aug 25 2000

More terms from Paul D. Hanna, Feb 26 2005

A278414 T(n,k)=Number of nXk 0..2 arrays with rows and columns in lexicographic nondecreasing order but with exactly two mistakes.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Nov 21 2016

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

			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

R. H. Hardin, Table of n, a(n) for n = 1..127

Column 1 is A005716(n+1).

Empirical for column k:
k=1: [polynomial of degree 8]
k=2: [polynomial of degree 26]
k=3: [polynomial of degree 80]

A278634 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.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Nov 24 2016

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

			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

R. H. Hardin, Table of n, a(n) for n = 1..127

Column 1 is A005716(n+1).

Empirical for column k:
k=1: [polynomial of degree 8]
k=2: [polynomial of degree 24]
k=3: [polynomial of degree 70]

A005715 Coefficient of x^7 in expansion of (1+x+x^2)^n.

Original entry on oeis.org

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

Offset: 4

N. J. A. Sloane

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).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
R. K. Guy, Letter to N. J. A. Sloane, 1987
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Trinomial Coefficient
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000574, A005581, A005712, A005714, A005716, A111808.

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

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[8]: n in [4..33] ]; // Bruno Berselli, Jun 17 2012

A005715:=(z-2)*(z**2-2)/(z-1)**8; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005715 := n -> GegenbauerC(`if`(7A005715(n)), n=4..20); # Peter Luschny, May 10 2016

CoefficientList[Series[(x-2)*(x^2-2)/(1-x)^8,{x,0,40}],x] (* Vincenzo Librandi, Jun 16 2012 *)

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) = A111808(n,7) for n>6. - Reinhard Zumkeller, Aug 17 2005
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) = 4*binomial(n,4) + 10*binomial(n,5) + 6*binomial(n,6) + binomial(n,7) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 7 if 7Peter Luschny, May 10 2016

More terms from Vladeta Jovovic, Oct 02 2000

A064054 Tenth column of trinomial coefficients.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Aug 29 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

A005716 (ninth column), A111808.

A064054 := n -> GegenbauerC(`if`(9A064054(n)), n=5..20); # Peter Luschny, May 10 2016

Table[GegenbauerC[9, -n, -1/2], {n,5,50}] (* G. C. Greubel, Feb 28 2017 *)

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

a(n) = A027907(n+5, 9).
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) = A111808(n+5,9) for n>3. - Reinhard Zumkeller, Aug 17 2005
a(n) = 5*binomial(n+5,5) + 20*binomial(n+5,6) + 21*binomial(n+5,7) + 8*binomial(n+5,8) + binomial(n+5,9) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 9 if 9Peter Luschny, May 10 2016

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cp's OEIS Frontend

A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

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A005712 Coefficient of x^4 in expansion of (1+x+x^2)^n.

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A111808 Left half of trinomial triangle (A027907), triangle read by rows.

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A000574 Coefficient of x^5 in expansion of (1 + x + x^2)^n.

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A005714 Coefficient of x^6 in expansion of (1+x+x^2)^n.

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A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

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