A026729 -id:A026729 - cp's OEIS frontend

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

1, 0, 1, 0, -1, 1, 0, 2, -2, 1, 0, -5, 5, -3, 1, 0, 14, -14, 9, -4, 1, 0, -42, 42, -28, 14, -5, 1, 0, 132, -132, 90, -48, 20, -6, 1, 0, -429, 429, -297, 165, -75, 27, -7, 1, 0, 1430, -1430, 1001, -572, 275, -110, 35, -8, 1, 0, -4862, 4862, -3432, 2002, -1001, 429, -154, 44, -9, 1, 0, 16796, -16796, 11934, -7072, 3640, -1638

Offset: 0

Paul Barry, Sep 23 2004

Row sums are generalized Catalan numbers A064310. Diagonal sums are 0^n+(-1)^n*A030238(n-2). Inverse is A026729, as number triangle. Columns have g.f. (xc(-x))^k=((sqrt(1+4x)-1)/2)^k.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938. - Philippe Deléham, May 31 2005

			Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1}, ...
Triangle begins
  1;
  0,    1;
  0,   -1,    1;
  0,    2,   -2,   1;
  0,   -5,    5,  -3,    1;
  0,   14,  -14,   9,   -4,   1;
  0,  -42,   42, -28,   14,  -5,  1;
  0,  132, -132,  90,  -48,  20, -6,  1;
  0, -429,  429, -297, 165, -75, 27, -7, 1;
Production matrix is
  0,  1,
  0, -1,  1,
  0,  1, -1,  1,
  0, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1, -1,  1

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Cf. A033184, A000108.
Cf. A106566 (unsigned version), A059365
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

T[n_, k_]:= If[n == 0 && k == 0, 1, If[n == 0 && k > 0, 0, (-1)^(n + k)*Binomial[2*n - k - 1, n - k]*k/n]];  Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

{T(n,k) = if(n == 0 && k == 0, 1, if(n == 0 && k > 0, 0, (-1)^(n + k)*binomial(2*n - k - 1, n - k)*k/n))};
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 31 2017

T(n, k) = (-1)^(n+k)*binomial(2*n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0. - Philippe Deléham, May 31 2005

A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 0

Paul Barry, Apr 29 2005

A skew version of Sierpinski’s triangle A047999. - Johannes W. Meijer, Jun 05 2011
Row sums are A002487(n+1). Diagonal sums are A106345. Inverse is A106346.
Triangle formed by reading T triangle mod 2 with T := A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167. - Philippe Deléham, Dec 18 2008

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 0, 1;

G. C. Greubel, Rows n = 0..50, flattened
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (1.6) p. 2.

Cf. A047999, A002487.
Cf. A106345 (diagonal sums), A106346 (inverse).
Cf. A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167.

Flat(List([0..15], n-> List([0..n], k-> (Binomial(k,n-k) mod 2) ))); # G. C. Greubel, Feb 07 2020

[ Binomial(k,n-k) mod 2: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 07 2020

seq(seq(`mod`(binomial(k, n-k), 2), k = 0..n), n = 0..15); # G. C. Greubel, Feb 07 2020

Table[Mod[Binomial[k, n-k], 2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 18 2017 *)

T(n,k) = binomial(k,n-k)%2;
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020

[[ mod(binomial(k,n-k), 2) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Feb 07 2020

A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 5, 8, 0, 0, 6, 15, 17, 16, 0, 0, 24, 62, 68, 49, 32, 0, 0, 120, 322, 359, 243, 129, 64, 0, 0, 720, 2004, 2308, 1553, 756, 321, 128, 0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256, 0, 0, 40320, 119664

Offset: 0

Philippe Deléham, Oct 19 2005

Let R(m,n,k), 0<=k<=n, the Riordan array (1, x*g(x)) where g(x) is g.f. of the m-fold factorials . Then Sum_{k, 0<=k<=n} = R(m,n,k) = Sum_{k, 0<=k<=n} T(n,k)*m^(n-k).
For m = -1, R(-1,n,k) is A026729(n,k).
For m = 0, R(0,n,k) is A097805(n,k).
For m = 1, R(1,n,k) is A084938(n,k).
For m = 2, R(2,n,k) is A111106(n,k).

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 4;
.0, 0, 2, 5, 8;
.0, 0, 6, 15, 17, 16;
.0, 0, 24, 62, 68, 49, 32;
.0, 0, 120, 322, 359, 243, 129, 64;
.0, 0, 720, 2004, 2308, 1553, 756, 321, 128;
.0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256;
.0, 0, 40320, 119664, 148232, 105048, 49840, 18066, 5756, 1793, 512;
....................................................................
At y=2: Sum_{k=0..n} 2^k*T(n,k) = A113327(n) where (1 + 2*x + 8*x^2 + 36*x^3 +...+ A113327(n)*x^n +..) = 1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 +..) ).
At y=3: Sum_{k=0..n} 3^k*T(n,k) = A113328(n) where (1 + 3*x + 18*x^2 + 117*x^3 +...+ A113328(n)*x^n +..) = 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 +..) ).
At y=4: Sum_{k=0..n} 4^k*T(n,k) = A113329(n) where (1 + 4*x + 32*x^2 + 272*x^3 +...+ A113329(n)*x^n +..) = 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 +..) ).

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000045, A026729, A051295, A084938, A097805, A111106, A112934.
Cf. m-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542.
Cf. A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

T[n_, k_] := Module[{x = X + X*O[X]^n, y = Y + Y*O[Y]^k}, A = 1/(1 - x*y*Sum[x^j*Product[y + i, {i, 0, j - 1}], {j, 0, n}]); Coefficient[ Coefficient[A, X, n], Y, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, from PARI *)

{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); A=1/(1-x*y*sum(j=0,n,x^j*prod(i=0,j-1,y+i))); return(polcoeff(polcoeff(A,n,X),k,Y))} (Hanna)

Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n, k) = A000045(n+1), Fibonacci numbers.
Sum_{k, 0<=k<=n} T(n, k) = A051295(n).
Sum_{k, 0<=k<=n} 2^(n-k)*T(n, k) = A112934(n).
T(0, 0) = 1, T(n, n) = 2^(n-1).
G.f.: A(x, y) = 1/(1 - x*y*Sum_{j>=0} (y-1+j)!/(y-1)!*x^j ). - Paul D. Hanna, Oct 26 2005

A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 0, 1, 12, 55

Offset: 0

N. J. A. Sloane, Mar 17 2000

Another version of Pascal's triangle A007318.
As a triangle read by rows, it is (1,0,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938 and it is the Riordan array (1/(1-x), x^2/(1-x)). The row sums of this triangle are F(n+1) = A000045(n+1). - Philippe Deléham, Dec 11 2011
As a triangle, binomial(n-k, k) is also the number of ways to add k pierced circles to a path graph P_n so that no two circles share a vertex (see Lemma 3.1 at page 5 in Owad and Tsvietkova). - Stefano Spezia, May 18 2022
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, ...
  1, 5, 10, 10, 5, 1, ...
As a triangle, this begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  0, 0;
  1, 3,  1, 0, 0;
  1, 4,  3, 0, 0, 0;
  1, 5,  6, 1, 0, 0, 0;
  1, 6, 10, 4, 0, 0, 0, 0;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5459
Math Stack Exchange, Homology of the n-torus using the Künneth Formula
Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.
Index entries for triangles and arrays related to Pascal's triangle

The official entry for Pascal's triangle is A007318. See also A026729 (the same array read by downward antidiagonals).
As a triangle without zeros: A011973.
Cf. A052509, A054123, A054124, A008949.

/* As triangle */ [[Binomial(n-k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 08 2017

with(combinat): for s from 0 to 20 do for n from s to 0 by -1 do printf(`%d,`, binomial(n, s-n)) od:od: # James Sellers, Mar 17 2000

Flatten[ Table[ Binomial[n-k , k], {n, 0, 13}, {k, 0, n}]]  (* Jean-François Alcover, Dec 05 2012 *)

T(n,k) = binomial(n,k) \\ Charles R Greathouse IV, Feb 07 2017

As a triangle: T(n,k) = A026729(n,n-k).

G.f. of the triangular version: 1/(1-x-x^2*y). - R. J. Mathar, Aug 11 2015

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

A127782 G.f. satisfies A(x) = 1 + x*A(x+x^2).

Original entry on oeis.org

1, 1, 1, 2, 4, 11, 33, 114, 438, 1845, 8458, 41823, 221539, 1250269, 7481758, 47278652, 314374316, 2192798077, 16000160519, 121831654450, 965946444587, 7958739329386, 68023023892680, 602115897105136, 5511499584735858

Offset: 0

Paul D. Hanna, Jan 28 2007

nonn

Equals eigensequence of triangle A026729. - Gary W. Adamson, Jan 16 2009

In Barry[2011] on page 9 is Example 12 where the first column of the eigentriangle of the skew binomial matrix is this sequence. - Michael Somos, Oct 03 2024

			G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 33*x^6 + 114*x^7 + ... - _Michael Somos_, Oct 03 2024

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Cf. A026729, A000110, A360885, A360886, A360894.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*binomial(n-i, i-1), i=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

nmax = 20; b = ConstantArray[0, nmax]; b[[1]] = 1; Do[b[[n+2]] = Sum[Binomial[n-k, k]*b[[n-k+1]], {k, 0, n}], {n, 0, nmax-2}]; b (* Vaclav Kotesovec, Mar 12 2014 *)
a[ n_] := If[n<1, Boole[n==0], a[n] = Sum[Binomial[k, n-1-k] * a[k], {k, 0, n-1}]]; (* Michael Somos, Oct 03 2024 *)

a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=1+x*subst(A,x,x+x^2)); polcoeff(A,n)

a(n)=if(n==0, 1, sum(k=0, (n-1)\2,binomial(n-1-k,k)*a(n-1-k))); \\ corrected by Seiichi Manyama, Feb 25 2023

{a(n) = my(A = 1 + O(x)); for(k=1, n, A = 1 + x*subst(A, x, x+x^2)); polcoeff(A, n)}; /* Michael Somos, Oct 03 2024 */

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1-k,k) * a(n-1-k) for n>0 with a(0)=1. [corrected by Seiichi Manyama, Feb 25 2023]
a(n) ~ c * Bell(n) * LambertW(n) / (n*exp(LambertW(n)^2/2)), where c = 1.93210869..., or a(n) ~ c * exp(n/LambertW(n) - LambertW(n)^2/2 - 1 - n) * n^(n-1) / (LambertW(n)^(n-1) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Mar 12 2014
a(n) = Sum_{k=0..n-1} binomial(k, n-k-1) * a(k) for n>0 with a(0)=1. (from Barry[2011]) - Michael Somos, Oct 03 2024

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

A213887 Triangle of coefficients of representations of columns of A213743 in binomial basis.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 0, 4, 6, 4, 1, 0, 0, 3, 10, 10, 5, 1, 0, 0, 2, 12, 20, 15, 6, 1, 0, 0, 1, 12, 31, 35, 21, 7, 1, 0, 0, 0, 10, 40, 65, 56, 28, 8, 1, 0, 0, 0, 6, 44, 101, 120, 84, 36, 9, 1, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 23 2012

This triangle is the third array in the sequence of arrays A026729, A071675 considered as triangles.
Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th row of the triangle. 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 A213743. For example, s_1(n)=binomial(n,1)=n is the first column of A213743 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213743 for n>1, etc. In particular (see comment in A213743), in cases k=6,7,8,9  s_k(n) is A064056(n+2), A064057(n+2), A064058(n+2), A000575(n+3) respectively.
Riordan array (1,x+x^2+x^3+x^4). A186332 with additional 0 column. - Ralf Stephan, Dec 31 2013

			As a triangle, this begins
n/k.|..0....1....2....3....4....5....6....7....8....9
=====================================================
.0..|..1
.1..|..0....1
.2..|..0....1....1
.3..|..0....1....2....1
.4..|..0....1....3....3....1
.5..|..0....0....4....6....4....1
.6..|..0....0....3...10...10....5....1
.7..|..0....0....2...12...20...15....6....1
.8..|..0....0....1...12...31...35...21....7....1
.9..|..0....0....0...10...40...65...56...28....8....1

Cf. A026729, A071675, A030528 (parts <=2), A078803 (parts <=3), A213888 (parts <=5), A061676 and A213889 (parts <=6).

pts := 4; # A213887
g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
for n from 0 to 13 do
    for k from 0 to n do
        coeftayl(g,z=0,n) ;
        coeftayl(%,t=0,k) ;
        printf("%d ",%) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 28 2025

A213888 Triangle of coefficients of representations of columns of A213744 in binomial basis.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 0, 5, 10, 10, 5, 1, 0, 0, 4, 15, 20, 15, 6, 1, 0, 0, 3, 18, 35, 35, 21, 7, 1, 0, 0, 2, 19, 52, 70, 56, 28, 8, 1, 0, 0, 1, 18, 68, 121, 126, 84, 36, 9, 1, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 23 2012

This triangle is the fourth array in the sequence of arrays A026729, A071675, A213887,..., such that the first two arrays are considered as triangles.

Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th row of the triangle. 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 A213744. For example, s_1(n)=binomial(n,1)=n is the first column of A213744 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213744 for n>1, etc. In particular (see comment inA213744), in cases k=7,8,9 s_k(n) is A063262(n+2), A063263(n+2), A063264(n+2) respectively.

			As a triangle, this begins
n/k.|..0....1....2....3....4....5....6....7....8....9
=====================================================
.0..|..1
.1..|..0....1
.2..|..0....1....1
.3..|..0....1....2....1
.4..|..0....1....3....3....1
.5..|..0....1....4....6....4....1
.6..|..0....0....5...10...10....5....1
.7..|..0....0....4...15...20...15....6....1
.8..|..0....0....3...18...35...35...21....7....1
.9..|..0....0....2...19...52...70...56...28....8....1

Cf. A026729, A071675, A213887, A030528 (parts <=2), A078803 (parts <=3), A213887 (parts <=4).

pts := 5; # A213888
g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
for n from 0 to 13 do
    for k from 0 to n do
        coeftayl(g,z=0,n) ;
        coeftayl(%,t=0,k) ;
        printf("%d ",%) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 28 2025

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

cp's OEIS Frontend

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

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A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

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A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

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A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

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

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A127782 G.f. satisfies A(x) = 1 + x*A(x+x^2).

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