A005715 -id:A005715 - 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

A005716 Coefficient of x^8 in expansion of (1+x+x^2)^n.

Original entry on oeis.org

1, 15, 90, 357, 1107, 2907, 6765, 14355, 28314, 52624, 93093, 157950, 258570, 410346, 633726, 955434, 1409895, 2040885, 2903428, 4065963, 5612805, 7646925, 10293075, 13701285, 18050760, 23554206, 30462615, 39070540, 49721892, 62816292, 78816012, 98253540

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 (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000574, A005581, A005712, A005714, A005715, A111808.

I:=[1, 15, 90, 357, 1107, 2907, 6765, 14355, 28314]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, Jun 16 2012

/* By definition: */ P:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2)^n)[9]: n in [4..32] ]; // Bruno Berselli, Jun 17 2012

A005716:=-(6*z-9*z**2+3*z**3+1)/(z-1)**9; # Conjectured by Simon Plouffe in his 1992 dissertation.
A005716 := n -> GegenbauerC(`if`(8A005716(n)), n=4..20); # Peter Luschny, May 10 2016

CoefficientList[Series[(1+6*x-9*x^2+3*x^3)/(1-x)^9,{x,0,40}],x] (* Vincenzo Librandi, Jun 16 2012 *)

a(n) = binomial(n+1, 5)*(n^2+23*n-84)*(n+10)/336, n >= 4.
G.f.: (x^4)*(1+6*x-9*x^2+3*x^3)/(1-x)^9. (Numerator polynomial is N3(8, x) from A063420).
a(n) = A027907(n, 8), n >= 4 (ninth column of trinomial coefficients).
a(n) = A111808(n,8) for n>7. - Reinhard Zumkeller, Aug 17 2005
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). Vincenzo Librandi, Jun 16 2012
a(n) = binomial(n,4) + 10*binomial(n,5) + 15*binomial(n,6) + 7*binomial(n,7) + binomial(n,8) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 8 if 8Peter Luschny, May 10 2016

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

A027778 a(n) = 5(n+1)binomial(n+2, 5)/2.

Original entry on oeis.org

10, 75, 315, 980, 2520, 5670, 11550, 21780, 38610, 65065, 105105, 163800, 247520, 364140, 523260, 736440, 1017450, 1382535, 1850695, 2443980, 3187800, 4111250, 5247450, 6633900, 8312850, 10331685, 12743325, 15606640, 18986880, 22956120, 27593720, 32986800, 39230730

Offset: 3

Thi Ngoc Dinh (via R. K. Guy)

Number of 8-subsequences of [ 1, n ] with just 2 contiguous pairs.

Michael De Vlieger, Table of n, a(n) for n = 3..10000
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Journal of Integer Sequences, Vol. 24 (2021), Article 21.8.6; ResearchGate link.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Not equal to 5*A005715(n+1)/2.

DeleteCases[CoefficientList[Series[5 x^3*(2 + x)/(1 - x)^7, {x, 0, 24}], x], 0] (* Michael De Vlieger, Jul 16 2021 *)

G.f.: 5*x^3*(2+x)/(1-x)^7.
a(n) = binomial(n+1, 4)*binomial(n+2, 2). - Zerinvary Lajos, Apr 28 2005, corrected by R. J. Mathar, Feb 13 2016
From Amiram Eldar, Feb 01 2022: (Start)
Sum_{n>=3} 1/a(n) = 239/18 - 4*Pi^2/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = 2*Pi^2/3 + 64*log(2)/3 - 383/18. (End)

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

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

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A005716 Coefficient of x^8 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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A027778 a(n) = 5*(n+1)*binomial(n+2, 5)/2.

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A027778 a(n) = 5(n+1)binomial(n+2, 5)/2.