A001288 -id:A001288 - cp's OEIS frontend

A135278 Triangle read by rows, giving the numbers T(n,m) = binomial(n+1, m+1); or, Pascal's triangle A007318 with its left-hand edge removed.

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7, 1, 8, 28, 56, 70, 56, 28, 8, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 12, 66, 220, 495, 792, 924, 792

Offset: 0

Zerinvary Lajos, Dec 02 2007

T(n,m) is the number of m-faces of a regular n-simplex.
An n-simplex is the n-dimensional analog of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher, i.e., a set of points such that no m-plane contains more than (m + 1) of them. Such points are said to be in general position.
Reversing the rows gives A074909, which as a linear sequence is essentially the same as this.
From Tom Copeland, Dec 07 2007: (Start)
T(n,k) * (k+1)! = A068424. The comment on permuted words in A068424 shows that T is related to combinations of letters defined by connectivity of regular polytope simplexes.
If T is the diagonally-shifted Pascal matrix, binomial(n+m, k+m), for m=1, then T is a fundamental type of matrix that is discussed in A133314 and the following hold.
The infinitesimal matrix generator is given by A132681, so T = LM(1) of A132681 with inverse LM(-1).
With a(k) = (-x)^k / k!, T * a = [ Laguerre(n,x,1) ], a vector array with index n for the Laguerre polynomials of order 1. Other formulas for the action of T are given in A132681.
T(n,k) = (1/n!) (D_x)^n (D_t)^k Gf(x,t) evaluated at x=t=0 with Gf(x,t) = exp[ t * x/(1-x) ] / (1-x)^2.
[O.g.f. for T ] = 1 / { [ 1 - t * x/(1-x) ] * (1-x)^2 }. [ O.g.f. for row sums ] = 1 / { (1-x) * (1-2x) }, giving A000225 (without a leading zero) for the row sums. Alternating sign row sums are all 1. [Sign correction noted by Vincent J. Matsko, Jul 19 2015]
O.g.f. for row polynomials = [ (1+q)**(n+1) - 1 ] / [ (1+q) -1 ] = A(1,n+1,q) on page 15 of reference on Grassmann cells in A008292. (End)
Given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. The e.g.f. for the row polynomials of A is {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. - Tom Copeland, Aug 21 2008
A007318*A097806 as infinite lower triangular matrices. - Philippe Deléham, Feb 08 2009
Riordan array (1/(1-x)^2, x/(1-x)). - Philippe Deléham, Feb 22 2012
The elements of the matrix inverse are T^(-1)(n,k)=(-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 12 2013
Relation to K-theory: T acting on the column vector (-0,d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168 and also A104712, A111492, and A238363. - Tom Copeland, Apr 11 2014
Number of walks of length p>0 between any two distinct vertices of the complete graph K_(n+2) is W(n+2,p)=(-1)^(p-1)*Sum_{k=0..p-1} T(p-1,k)*(-n-2)^k = ((n+1)^p - (-1)^p)/(n+2) = (-1)^(p-1)*Sum_{k=0..p-1} (-n-1)^k. This is equal to (-1)^(p-1)*Phi(p,-n-1), where Phi is the cyclotomic polynomial when p is an odd prime. For K_3, see A001045; for K_4, A015518; for K_5, A015521; for K_6, A015531; for K_7, A015540. - Tom Copeland, Apr 14 2014
Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-1)^0 + A_1*(x-1)^1 + A_2*(x-1)^2 + ... + A_n*(x-1)^n. This sequence gives A_0, ..., A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 14 2014
See A074909 for associations among this array, the Bernoulli polynomials and their umbral compositional inverses, and the face polynomials of permutahedra and their duals (cf. A019538). - Tom Copeland, Nov 14 2014
From Wolfdieter Lang, Dec 10 2015: (Start)
A(r, n) = T(n+r-2, r-1) = risefac(n,r)/r! = binomial(n+r-1, r), for n >= 1 and r >= 1, gives the array with the number of independent components of a symmetric tensors of rank r (number of indices) and dimension n (indices run from 1 to n). Here risefac(n, k) is the rising factorial.
As(r, n) = T(n+1, r+1) = fallfac(n, r)/r! = binomial(n, r), r >= 1 and n >= 1 (with the triangle entries T(n, k) = 0 for n < k) gives the array with the number of independent components of an antisymmetric tensor of rank r and dimension n. Here fallfac is the falling factorial. (End)
The h-vectors associated to these f-vectors are given by A000012 regarded as a lower triangular matrix. Read as bivariate polynomials, the h-polynomials are the complete homogeneous symmetric polynomials in two variables, found in the compositional inverse of an e.g.f. for A008292, the h-vectors of the permutahedra. - Tom Copeland, Jan 10 2017
For a correlation between the states of a quantum system and the combinatorics of the n-simplex, see Boya and Dixit. - Tom Copeland, Jul 24 2017

			The triangle T(n, k) begins:
   n\k  0  1   2   3   4   5   6   7   8  9 10 11 ...
   0:   1
   1:   2  1
   2:   3  3   1
   3:   4  6   4   1
   4:   5 10  10   5   1
   5:   6 15  20  15   6   1
   6:   7 21  35  35  21   7   1
   7:   8 28  56  70  56  28   8   1
   8:   9 36  84 126 126  84  36   9   1
   9:  10 45 120 210 252 210 120  45  10  1
  10:  11 55 165 330 462 462 330 165  55 11  1
  11:  12 66 220 495 792 924 792 495 220 66 12  1
  ... reformatted by _Wolfdieter Lang_, Mar 23 2015
Production matrix begins
   2   1
  -1   1   1
   1   0   1   1
  -1   0   0   1   1
   1   0   0   0   1   1
  -1   0   0   0   0   1   1
   1   0   0   0   0   0   1   1
  -1   0   0   0   0   0   0   1   1
   1   0   0   0   0   0   0   0   1   1
- _Philippe Deléham_, Jan 29 2014
From _Wolfdieter Lang_, Nov 08 2018: (Start)
Recurrence [_Philippe Deléham_]: T(7, 3) = 2*35 + 35 - 15 - 20 = 70.
Recurrence from Riordan A- and Z-sequences: [1,1,repeat(0)] and [2, repeat(-1, +1)]: From Z: T(5, 0) = 2*5 - 10 + 10 - 5 + 1 = 6. From A: T(7, 3) = 35 + 35 = 70.
Boas-Buck column k=3 recurrence: T(7, 3) = (5/4)*(1 + 5 + 15 + 35) = 70. (End)

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
L. Boya and K. Dixit, Geometry of density states, arXiv:808.1930 [quant-phy], 2017.
V. Buchstaber, Lectures on Toric Topology, Trends in Mathematics - New Series, Information Center for Mathematical Sciences, Vol. 10, No. 1, 2008. p. 7.
Tom Copeland, Cyclotomic polynomials in combinatorics
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator Part VI on simplices
D. Dugger, A Geometric Introduction to K-Theory [From _Tom Copeland_, Apr 11 2014]
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 12.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 4.
B. Grünbaum and G. C. Shephard, Convex polytopes, Bull. London Math. Soc. (1969) 1 (3): 257-300.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4 (Added by _Tom Copeland_, Oct 01 2015).
Justin Hughes, Representations Arising from an Action on D-neighborhoods of Cayley Graphs, 2013; slides from a talk.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Wikipedia, Simplex

Cf. A007318, A014410, A228196.
Cf. Column sequences: A000027, A000217, A000292, A000332, A000389, A000579 - A000582, A001287, A001288, A010965 - A011001, A017713 - A017764.
Cf. A000012, A008292.

for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i) od;

Flatten[Table[CoefficientList[D[1/x ((x + 1) Exp[(x + 1) z] - Exp[z]), {z, k}] /. z -> 0, x], {k, 0, 11}]]
CoefficientList[CoefficientList[Series[1/((1 - x)*(1 - x - x*y)), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0, 20, for(k=0, n, print1(1/k!*sum(i=0, n, (prod(j=0, k-1, i-j))), ", "))) \\ Derek Orr, Oct 14 2014

Trow = lambda n: sum((x+1)^j for j in (0..n)).list()
for n in (0..10): print(Trow(n)) # Peter Luschny, Jul 09 2019

T(n, k) = Sum_{j=k..n} binomial(j,k) = binomial(n+1, k+1), n >= k >= 0, else 0. (Partial sum of column k of A007318 (Pascal), or summation on the upper binomial index (Graham et al. (GKP), eq. (5.10). For the GKP reference see A007318.) - Wolfdieter Lang, Aug 22 2012
E.g.f.: 1/x*((1 + x)*exp(t*(1 + x)) - exp(t)) = 1 + (2 + x)*t + (3 + 3*x + x^2)*t^2/2! + .... The infinitesimal generator for this triangle has the sequence [2,3,4,...] on the main subdiagonal and 0's elsewhere. - Peter Bala, Jul 16 2013
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 27 2013
T(n,k) = A193862(n,k)/2^k. - Philippe Deléham, Jan 29 2014
G.f.: 1/((1-x)*(1-x-x*y)). - Philippe Deléham, Mar 13 2014
From Tom Copeland, Mar 26 2014: (Start)
[From Copeland's 2007 and 2008 comments]
A) O.g.f.: 1 / { [ 1 - t * x/(1-x) ] * (1-x)^2 } (same as Deleham's).
B) The infinitesimal generator for T is given in A132681 with m=1 (same as Bala's), which makes connections to the ubiquitous associated Laguerre polynomials of integer orders, for this case the Laguerre polynomials of order one L(n,-t,1).
C) O.g.f. of row e.g.f.s: Sum_{n>=0} L(n,-t,1) x^n = exp[t*x/(1-x)]/(1-x)^2 = 1 + (2+t)x + (3+3*t+t^2/2!)x^2 + (4+6*t+4*t^2/2!+t^3/3!)x^3+ ... .
D) E.g.f. of row o.g.f.s: ((1+t)*exp((1+t)*x)-exp(x))/t (same as Bala's).
E) E.g.f. for T(n,k)*a(n-k): {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. For example, for a(k)=2^k, the e.g.f. for the row o.g.f.s is {(2+t) exp[(2+t)x] - 2 exp(2x)}/t.
(End)
From Tom Copeland, Apr 28 2014: (Start)
With different indexing
A) O.g.f. by row: [(1+t)^n-1]/t.
B) O.g.f. of row o.g.f.s: {1/[1-(1+t)*x] - 1/(1-x)}/t.
C) E.g.f. of row o.g.f.s: {exp[(1+t)*x]-exp(x)}/t.
These generating functions are related to row e.g.f.s of A111492. (End)
From Tom Copeland, Sep 17 2014: (Start)
A) U(x,s,t)= x^2/[(1-t*x)(1-(s+t)x)] = Sum_{n >= 0} F(n,s,t)x^(n+2) is a generating function for bivariate row polynomials of T, e.g., F(2,s,t)= s^2 + 3s*t + 3t^2 (Buchstaber, 2008).
B) dU/dt=x^2 dU/dx with U(x,s,0)= x^2/(1-s*x) (Buchstaber, 2008).
C) U(x,s,t) = exp(t*x^2*d/dx)U(x,s,0) = U(x/(1-t*x),s,0).
D) U(x,s,t) = Sum[n >= 0, (t*x)^n L(n,-:xD:,-1)] U(x,s,0), where (:xD:)^k=x^k*(d/dx)^k and L(n,x,-1) are the Laguerre polynomials of order -1, related to normalized Lah numbers. (End)
E.g.f. satisfies the differential equation d/dt(e.g.f.(x,t)) = (x+1)*e.g.f.(x,t) + exp(t). - Vincent J. Matsko, Jul 18 2015
The e.g.f. of the Norlund generalized Bernoulli (Appell) polynomials of order m, NB(n,x;m), is given by exponentiation of the e.g.f. of the Bernoulli numbers, i.e., multiple binomial self-convolutions of the Bernoulli numbers, through the e.g.f. exp[NB(.,x;m)t] = (t/(e^t - 1))^(m+1) * e^(xt). Norlund gave the relation to the factorials (x-1)!/(x-1-n)! = (x-1) ... (x-n) = NB(n,x;n), so T(n,m) = NB(m+1,n+2;m+1)/(m+1)!. - Tom Copeland, Oct 01 2015
From Wolfdieter Lang, Nov 08 2018: (Start)
Recurrences from the A- and Z- sequences for the Riordan triangle (see the W. Lang link under A006232 with references), which are A(n) = A019590(n+1), [1, 1, repeat (0)] and Z(n) = (-1)^(n+1)*A054977(n), [2, repeat(-1, 1)]:
  T(0, 0) = 1, T(n, k) = 0 for n < k, and T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1, and T(n, k) = T(n-1, k-1) + T(n-1, k), for n >= m >= 1.
Boas-Buck recurrence for columns (see the Aug 10 2017 remark in A036521 also for references):
  T(n, k) = ((2 + k)/(n - k))*Sum_{j=k..n-1} T(j, k), for n >= 1, k = 0, 1, ..., n-1, and input T(n, n) = 1, for n >= 0, (the BB-sequences are alpha(n) = 2 and beta(n) = 1). (End)
T(n, k) = [x^k] Sum_{j=0..n} (x+1)^j. - Peter Luschny, Jul 09 2019

Edited by Tom Copeland and N. J. A. Sloane, Dec 11 2007

A078812 Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 10, 6, 1, 5, 20, 21, 8, 1, 6, 35, 56, 36, 10, 1, 7, 56, 126, 120, 55, 12, 1, 8, 84, 252, 330, 220, 78, 14, 1, 9, 120, 462, 792, 715, 364, 105, 16, 1, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1, 11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20, 1

Offset: 0

Michael Somos, Dec 05 2002

Warning: formulas and programs sometimes refer to offset 0 and sometimes to offset 1.
Apart from signs, identical to A053122.
Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. - Philippe Deléham, Feb 16 2004
T(n,k) is the number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2) = 10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - Emeric Deutsch, Apr 09 2005
T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Oct 02 2008
This sequence is jointly generated with A085478 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + (x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012
Concerning Kimberling's recursion relations, see A102426. - Tom Copeland, Jan 19 2016
Subtriangle of the triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 27 2012
From Wolfdieter Lang, Aug 30 2012: (Start)
With offset [0,0] the triangle with entries R(n,k) = T(n+1,k+1):= binomial(n+k+1, 2*k+1), n >= k >= 0, and zero otherwise, becomes the Riordan lower triangular convolution matrix R = (G(x)/x, G(x)) with G(x):=x/(1-x)^2 (o.g.f. of A000027). This means that the o.g.f. of column number k of R is (G(x)^(k+1))/x. This matrix R is the inverse of the signed Riordan lower triangular matrix A039598, called in a comment there S.
The Riordan matrix with entries R(n,k), just defined, provides the transition matrix between the sequence entry F(4*m*(n+1))/L(2*l), with m >= 0, for n=0,1,... and the sequence entries 5^k*F(2*m)^(2*k+1) for k = 0,1,...,n, with F=A000045 (Fibonacci) and L=A000032 (Lucas). Proof: from the inverse of the signed triangle Riordan matrix S used in a comment on A039598.
For the transition matrix R (T with offset [0,0]) defined above, row n=2: F(12*m) /L(2*m) = 3*5^0*F(2*m)^1 + 4*5^1*F(2*m)^3 + 1*5^2*F(2*m)^5, m >= 0. (End)
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
For 1 <= k <= n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01. - Milan Janjic, Dec 20 2016
The infinite sum (Sum_{i >= 0} (T(s+i,1+i) / 2^(s+2*i)) * zeta(s+1+2*i)) = 1 allows any zeta(s+1) to be expressed as a sum of rational multiples of zeta(s+1+2*i) having higher arguments. For example, zeta(3) can be expressed as a sum involving zeta(5), zeta(7), etc. The summation for each s >= 1 uses the s-th diagonal of the triangle. - Robert B Fowler, Feb 23 2022
The convolution triangle of the nonnegative integers. - Peter Luschny, Oct 07 2022

			Triangle begins, 1 <= k <= n:
                          1
                        2   1
                      3   4   1
                    4  10   6   1
                  5  20  21   8   1
                6  35  56  36  10   1
              7  56 126 120  55  12   1
            8  84 252 330 220  78  14   1
From _Peter Bala_, Feb 11 2025: (Start)
The array factorizes as an infinite product of lower triangular arrays:
  / 1               \    / 1              \ / 1              \ / 1             \
  | 2    1           |   | 2   1          | | 0  1           | | 0  1          |
  | 3    4   1       | = | 3   2   1      | | 0  2   1       | | 0  0  1       | ...
  | 4   10   6   1   |   | 4   3   2  1   | | 0  3   2  1    | | 0  0  2  1    |
  | 5   20  21   8  1|   | 5   4   3  2  1| | 0  4   3  2  1 | | 0  0  3  2  1 |
  |...               |   |...             | |...             | |...            |
Cf. A092276. (End)

T. D. Noe, Rows n = 0..50 of triangle, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Nour-Eddine Fahssi, On the combinatorics of exclusion in Haldane fractional statistics, arXiv:1808.00045 [cond-mat.stat-mech], 2018.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
A. Laradji, and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart. 43, no. 4, (2005) 359-370.

This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.
Row sums give A001906. With signs: A053122.
The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.
Cf. A128908, A053123, A049310, A119900, A085478, A029653, A106195.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019

a078812 n k = a078812_tabl !! n !! k
a078812_row n = a078812_tabl !! n
a078812_tabl = [1] : [2, 1] : f [1] [2, 1] where
   f us vs = ws : f vs ws where
     ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))
                      (us ++ [0, 0])
-- Reinhard Zumkeller, Dec 16 2013

/* As triangle */ [[Binomial(n+k-1, 2*k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 01 2018

for n from 1 to 11 do seq(binomial(n+k-1,2*k-1),k=1..n) od; # yields sequence in triangular form; Emeric Deutsch, Apr 09 2005
# Uses function PMatrix from A357368. Adds a row and column above and to the left.
PMatrix(10, n -> n); # Peter Luschny, Oct 07 2022

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A085478 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A078812 *) (* Clark Kimberling, Feb 25 2012 *)
(* Second program *)
Table[Binomial[n+k+1, 2*k+1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

T(n,m):=sum(binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)*binomial(n+1,m+k+1),k,0,n-m); /* Vladimir Kruchinin, Apr 13 2016 */

{T(n, k) = if( n<0, 0, binomial(n+k-1, 2*k-1))};

{T(n, k) = polcoeff( polcoeff( x*y / (1 - (2 + y) * x + x^2) + x * O(x^n), n), k)};

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A078812 = lambda n,k: T(k,n)
[[A078812(n,k) for k in (1..n)] for n in (1..8)] # Peter Luschny, Mar 12 2016

[[binomial(n+k+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f.: x*y / (1 - (2 + y)*x + x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.
From Philippe Deléham, Feb 16 2004: (Start)
If indexing begins at 0 we have
T(n,k) = (n+k+1)!/((n-k)!*(2k+1))!.
T(n,k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n < k.
T(n,k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n,k) = 0 if k < 0, T(0, 0)=1 and T(0, k) = 0 for k > 0.
G.f. for the column k: Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2).
Row sums: Sum_{k>=0} T(n, k) = A001906(n+1). (End)
Antidiagonal sums are A000079(n) = Sum_{k=0..floor(n/2)} binomial(n+k+1, n-k). - Paul Barry, Jun 21 2004
Riordan array (1/(1-x)^2, x/(1-x)^2). - Paul Barry, Oct 22 2006
T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + 2*T(n-1,k) - T(n-2,k). - Philippe Deléham, Jan 26 2010
For another version see A128908. - Philippe Deléham, Mar 27 2012
T(n,m) = Sum_{k=0..n-m} (binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)* binomial(n+1,m+k+1)). - Vladimir Kruchinin, Apr 13 2016
T(n, k) = T(n-1, k) + (T(n-1, k-1) + T(n-2, k-1) + T(n-3, k-1) + ...) for k >= 2 with T(n, 1) = n. - Peter Bala, Feb 11 2025
From Peter Bala, May 04 2025: (Start)
With the column offset starting at 0, the n-th row polynomial B(n, x) = 1/sqrt(x + 4) * Chebyshev_U(2*n+1, (1/2)*sqrt(x + 4)) = (-1)^n * Chebyshev_U(n, -(1/2)*(x + 2)).
B(n, x) / Product_{k = 1..2*n} (1 + 1/B(k, x)) = b(n, x), the n-th row polynomial of A085478. (End)

Edited by N. J. A. Sloane, Apr 28 2008

A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

Original entry on oeis.org

1, -2, 1, 3, -4, 1, -4, 10, -6, 1, 5, -20, 21, -8, 1, -6, 35, -56, 36, -10, 1, 7, -56, 126, -120, 55, -12, 1, -8, 84, -252, 330, -220, 78, -14, 1, 9, -120, 462, -792, 715, -364, 105, -16, 1, -10, 165, -792, 1716, -2002, 1365, -560, 136, -18, 1, 11, -220, 1287, -3432, 5005, -4368, 2380, -816, 171, -20

Offset: 0

Wolfdieter Lang

Apart from signs, identical to A078812.
Another version with row-leading 0's and differing signs is given by A285072.
G.f. for row polynomials S(n,x-2) (signed triangle): 1/(1+(2-x)*z+z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-(2+x)*z+z^2) for row polynomials.
Row sums (signed triangle) A049347(n) (periodic(1,-1,0)). Row sums (unsigned triangle) A001906(n+1)=F(2*(n+1)) (even-indexed Fibonacci).
In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The (unsigned) column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for m=0..5, resp. For m=6..23 they are A010966..(+2)..A011000 and for m=24..49 they are A017713..(+2)..A017763.
Riordan array (1/(1+x)^2,x/(1+x)^2). Inverse array is A039598. Diagonal sums have g.f. 1/(1+x^2). - Paul Barry, Mar 17 2005. Corrected by Wolfdieter Lang, Nov 13 2012.
Unsigned version is in A078812. - Philippe Deléham, Nov 05 2006
Also row n gives (except for an overall sign) coefficients of characteristic polynomial of the Cartan matrix for the root system A_n. - Roger L. Bagula, May 23 2007
From Wolfdieter Lang, Nov 13 2012: (Start)
The A-sequence for this Riordan triangle is A115141, and the Z-sequence is A115141(n+1), n>=0. For A- and Z-sequences for Riordan matrices see the W. Lang link under A006232 with details and references.
S(n,x^2-2) = sum(r(j,x^2),j=0..n) with Chebyshev's S-polynomials and r(j,x^2) := R(2*j+1,x)/x, where R(n,x) are the monic integer Chebyshv T-polynomials with coefficients given in A127672. Proof from comparing the o.g.f. of the partial sum of the r(j,x^2) polynomials (see a comment on the signed Riordan triangle A111125) with the present Riordan type o.g.f. for the row polynomials with x -> x^2.  (End)
S(n,x^2-2) = S(2*n+1,x)/x, n >= 0, from the odd part of the bisection of the o.g.f. - Wolfdieter Lang, Dec 17 2012
For a relation to a generator for the Narayana numbers A001263, see A119900, whose columns are unsigned shifted rows (or antidiagonals) of this array, referring to the tables in the example sections. - Tom Copeland, Oct 29 2014
The unsigned rows of this array are alternating rows of a mirrored A011973 and alternating shifted rows of A030528 for the Fibonacci polynomials. - Tom Copeland, Nov 04 2014
Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): a(n, m) =  (2*(m + 1)/(n - m))*Sum_{k = m..n-1} (-1)^(n-k)*a(k, m), with input a(n, n) = 1, and a(n,k) = 0 for n < k. - Wolfdieter Lang, Jun 03 2020
Row n gives the characteristic polynomial of the (n X n)-matrix M where M[i,j] = 2 if i = j, -1 if |i-j| = 1 and 0 otherwise. The matrix M is positive definite and has 2-condition number (cot(Pi/(2*n+2)))^2. - Jianing Song, Jun 21 2022
Also the convolution triangle of (-1)^(n+1)*n. - Peter Luschny, Oct 07 2022

			The triangle a(n,m) begins:
n\m   0    1    2     3     4     5     6    7    8  9
0:    1
1:   -2    1
2:    3   -4    1
3:   -4   10   -6     1
4:    5  -20   21    -8     1
5:   -6   35  -56    36   -10     1
6:    7  -56  126  -120    55   -12     1
7:   -8   84 -252   330  -220    78   -14    1
8:    9 -120  462  -792   715  -364   105  -16    1
9:  -10  165 -792  1716 -2002  1365  -560  136  -18  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 13 2012
E.g., fourth row (n=3) {-4,10,-6,1} corresponds to the polynomial S(3,x-2) = -4+10*x-6*x^2+x^3.
From _Wolfdieter Lang_, Nov 13 2012: (Start)
Recurrence: a(5,1) = 35 = 1*5 + (-2)*(-20) -1*(10).
Recurrence from Z-sequence [-2,-1,-2,-5,...]: a(5,0) = -6 = (-2)*5 + (-1)*(-20) + (-2)*21 + (-5)*(-8) + (-14)*1.
Recurrence from A-sequence [1,-2,-1,-2,-5,...]: a(5,1) = 35 = 1*5  + (-2)*(-20) + (-1)*21 + (-2)*(-8) + (-5)*1.
(End)
E.g., the fourth row (n=3) {-4,10,-6,1} corresponds also to the polynomial S(7,x)/x = -4 + 10*x^2 - 6*x^4 + x^6. - _Wolfdieter Lang_, Dec 17 2012
Boas-Buck type recurrence: -56 = a(5, 2) = 2*(-1*1 + 1*(-6) - 1*21) = -2*28 = -56. - _Wolfdieter Lang_, Jun 03 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62.
Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S.: ISBN 0-8218-2848-7, 1978, p. 463.

T. D. Noe, Rows n=0..50 of triangle, flattened
Wolfdieter Lang, First rows of the triangle.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group, J. of Integer Sequences, 8(2005), 1-16.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - _Tom Copeland_, Oct 11 2014
Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
Eric Weisstein's World of Mathematics, Cartan Matrix
Eric Weisstein's World of Mathematics, Dynkin Diagram
Index entries for sequences related to Chebyshev polynomials.

Cf. A005248, A127677, A053123, A049310.
Cf. A011973, A030528, A034867, A119900.
Cf. A285072 (version with row-leading 0's and differing signs). - Eric W. Weisstein, Apr 09 2017

seq(seq((-1)^(n+m)*binomial(n+m+1,2*m+1),m=0..n),n=0..10); # Robert Israel, Oct 15 2014
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> -(-1)^n*n); # Peter Luschny, Oct 07 2022

T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] (* Roger L. Bagula, May 23 2007 *)
(* Alternative code for the matrices from MathWorld: *)
sln[n_] := 2IdentityMatrix[n] - PadLeft[PadRight[IdentityMatrix[n - 1], {n, n - 1}], {n, n}] - PadLeft[PadRight[IdentityMatrix[n - 1], {n - 1, n}], {n, n}] (* Roger L. Bagula, May 23 2007 *)

@CachedFunction
def A053122(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A053122(n-1,k-1)-A053122(n-2,k)-2*A053122(n-1,k)
for n in (0..9): [A053122(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

a(n, m) := 0 if n

a(n, m) = -2*a(n-1, m) + a(n-1, m-1) - a(n-2, m), a(n, -1) := 0 =: a(-1, m), a(0, 0)=1, a(n, m) := 0 if n

O.g.f. for m-th column (signed triangle): ((x/(1+x)^2)^m)/(1+x)^2.

From Jianing Song, Jun 21 2022: (Start)

T(n,k) = [x^k]f_n(x), where f_{-1}(x) = 0, f_0(x) = 1, f_n(x) = (x-2)*f_{n-1}(x) - f_{n-2}(x) for n >= 2.

f_n(x) = (((x-2+sqrt(x^2-4*x))/2)^(n+1) - ((x-2-sqrt(x^2-4*x))/2)^(n+1))/sqrt(x^2-4x).

The roots of f_n(x) are 2 + 2*cos(k*Pi/(n+1)) = 4*(cos(k*Pi/(2*n+2)))^2 for 1 <= k <= n. (End)

A034867 Triangle of odd-numbered terms in rows of Pascal's triangle.

Original entry on oeis.org

1, 2, 3, 1, 4, 4, 5, 10, 1, 6, 20, 6, 7, 35, 21, 1, 8, 56, 56, 8, 9, 84, 126, 36, 1, 10, 120, 252, 120, 10, 11, 165, 462, 330, 55, 1, 12, 220, 792, 792, 220, 12, 13, 286, 1287, 1716, 715, 78, 1, 14, 364, 2002, 3432, 2002, 364, 14, 15, 455, 3003, 6435, 5005, 1365, 105, 1

Offset: 0

N. J. A. Sloane

Also triangle of numbers of n-sequences of 0,1 with k subsequences of consecutive 01 because this number is C(n+1,2*k+1). - Roger Cuculiere (cuculier(AT)imaginet.fr), Nov 16 2002
From Gary W. Adamson, Oct 17 2008: (Start)
Received from Herb Conn:
Let T = tan x, then
  tan x = T
  tan 2x = 2T / (1 - T^2)
  tan 3x = (3T - T^3) / (1 - 3T^2)
  tan 4x = (4T - 4T^3) / (1 - 6T^2 + T^4)
  tan 5x = (5T - 10T^3 + T^5) / (1 - 10T^2 + 5T^4)
  tan 6x = (6T - 20T^3 + 6T^5) / (1 - 15T^2 + 15T^4 - T^6)
  tan 7x = (7T - 35T^3 + 21T^5 - T^7) / (1 - 21T^2 + 35T^4 - 7T^6)
  tan 8x = (8T - 56T^3 + 56T^5 - 8T^7) / (1 - 28T^2 + 70T^4 - 28T^6 + T^8)
  tan 9x = (9T - 84T^3 + 126T^5 - 36T^7 + T^9) / (1 - 36 T^2 + 126T^4 - 84T^6 + 9T^8)
... To get the next one in the series, (tan 10x), for the numerator add:
  9....84....126....36....1 previous numerator +
  1....36....126....84....9 previous denominator =
  10..120....252...120...10 = new numerator
For the denominator add:
  ......9.....84...126...36...1 = previous numerator +
  1....36....126....84....9.... = previous denominator =
  1....45....210...210...45...1 = new denominator
...where numerators = A034867, denominators = A034839
(End)
Column k is the sum of columns 2k and 2k+1 of A007318. - Philippe Deléham, Nov 12 2008
Triangle, with zeros omitted, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
The row polynomials N(n,x) = Sum_{k=0..floor((n-1)/2)} T(n-1,k)*x^k, and D(n,x) = Sum_{k=0..floor(n/2)} A034839(n,k)*x^k, n >= 1, satisfy the recurrences N(n,x) = D(n-1,x) + N(n-1,x), D(n,x) = D(n-1,x) + x*N(n-1,x), with inputs N(1,x) = 1 = D(1,x). This is due to the Pascal triangle A007318 recurrence. Q(n,x) := tan(n*x)/tan(x) satisfies the recurrence Q(n,x) = (1 + Q(n-1,x))/(1 - v(x)*Q(n-1,x)) with input Q(1,x) = 1 and v = v(x) := (tan(x))^2. This recurrence is obtained from the addition theorem for tan(n*x) using n = 1 + (n-1). Therefore Q(n,x) = N(n,-v(x))/D(n,-v(x)). This proves the Gary W. Adamson contribution from above. See also A220673. This calculation was motivated by an e-mail of Thomas Olsen. The Oliver/Prodinger and Ma references resort to HAKEM Al Memo 239, Item 16, for the tan(n*x) formula in terms of tan(x). - Wolfdieter Lang, Jan 17 2013
The infinitesimal generator (infinigen) for the Narayana polynomials A090181/A001263 can be formed from the row polynomials P(n,y) of this entry. The resulting matrix is an instance of a matrix representation of the analytic infinigens presented in A145271 for general sets of binomial Sheffer polynomials and in A001263 and A119900 specifically for the Narayana polynomials. Given the column vector of row polynomials V = (1, P(1,x) = 2x, P(2,y) = 3x + x^2, P(3,y) = 4x + 4x^2, ...), form the lower triangular matrix M(n,k) = V(n-k,n-k), i.e., diagonally multiply the matrix with all ones on the diagonal and below by the components of V. Form the matrix MD by multiplying A132440^Transpose = A218272 = D (representing derivation of o.g.f.s) by M, i.e., MD = M*D. The non-vanishing component of the first row of (MD)^n * V / (n+1)! is the n-th Narayana polynomial. - Tom Copeland, Dec 09 2015
The diagonals of this entry are A078812 (also shifted A128908 and unsigned A053122, which are embedded in A030528, A102426, A098925, A109466, A092865). Equivalently, the antidiagonals of A078812 are the rows of A034867. - Tom Copeland, Dec 12 2015
Binomial(n,2k+1) is also the number of permutations avoiding both 132 and 213 with k peaks, i.e., positions with w[i]w[i+2]. - Lara Pudwell, Dec 19 2018
Binomial(n,2k+1) is also the number of permutations avoiding both 123 and 132 with k peaks, i.e., positions with w[i]w[i+2]. - Lara Pudwell, Dec 19 2018
The row polynomial P(n, x) = Sum_{0..floor(n/2)} T(n, k)*x^k appears as numerator polynomial of the diagonal sequence m of triangle A104698 as follows. G(m, x) = P(m, x^2)/(1 - x)^(m+1), for m >= 0. - Wolfdieter Lang, May 14 2025
Number of acyclic orientations of the path graph on n+1 vertices, with k-1 sinks. - Per W. Alexandersson, Aug 15 2025

			Triangle T starts:
  n\k   0   1   2   3   4  5 ...   ----------------------------------------
0:    1
1:    2
2:    3   1
3:    4   4
4:    5  10   1
5:    6  20   6
6:    7  35  21   1
7:    8  56  56   8
8:    9  84 126  36   1
9:   10 120 252 120  10
 10:   11 165 462 330  55  1
 11:   12 220 792 792 220 12
... ... reformatted and extended by - _Wolfdieter Lang_, May 14 2025

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 136.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, Counting pattern-avoiding permutations by big descents, arXiv:2408.15111 [math.CO], 2024. See p. 6.
S.-M. Ma, On some binomial coefficients related to the evaluation of tan(nx), arXiv preprint arXiv:1205.0735 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 13 2012
K. Oliver and H. Prodinger, The continued fraction expansion of Gauss' hypergeometric function and a new application to the tangent function, Transactions of the Royal Society of South Africa, Vol. 76 (2012), 151-154, [DOI], [PDF]. - From _N. J. A. Sloane_, Jan 03 2013
Eric Weisstein's World of Mathematics, Tangent. [From _Eric W. Weisstein_, Oct 18 2008]

Cf. A000129, A007318, A034839, A034867, A084938, A131980, A220673.
Cf. A001263, A090181, A119900, A132440, A145271.
Cf. A030528, A053122, A078812, A092865, A098925, A102426, A109466, A128908, A218272, A104698.
From Wolfdieter Lang, May 14 2025:(Start)
Row length A008619. Row sums A000079. Alternating row sums  A009545(n+1).
Column sequences (with certain offsets): A000027, A000292, A000389, A000580, A000582, A001288, ... (End)

/* as a triangle */ [[Binomial(n+1,2*k+1): k in [0..Floor(n/2)]]: n in [0..20]]; // G. C. Greubel, Mar 06 2018

seq(seq(binomial(n+1,2*k+1), k=0..floor(n/2)), n=0..14); # Emeric Deutsch, Apr 01 2005

u[1, x_] := 1; v[1, x_] := 1; z = 12;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]  (* A034839 as a triangle *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]  (* A034867 as a triangle *)
(* Clark Kimberling, Feb 18 2012 *)
Table[Binomial[n+1, 2*k+1], {n,0,20}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Mar 06 2018 *)

for(n=0,20, for(k=0,floor(n/2), print1(binomial(n+1,2*k+1), ", "))) \\ G. C. Greubel, Mar 06 2018

T(n,k) = C(n+1,2k+1) = Sum_{i=k..n-k} C(i,k) * C(n-i,k).
E.g.f.: 1+(exp(x)*sinh(x*sqrt(y)))/sqrt(y). - Vladeta Jovovic, Mar 20 2005
G.f.: 1/((1-z)^2-t*z^2). - Emeric Deutsch, Apr 01 2005
T(n,k) = Sum_{j = 0..n} A034839(j,k). - Philippe Deléham, May 18 2005
Pell(n+1) = A000129(n+1) = Sum_{k=0..n} T(n,k) * 2^k = (1/n!) Sum_{k=0..n} A131980(n,k) * 2^k. - Tom Copeland, Nov 30 2007
T(n,k) = A007318(n,2k) + A007318(n,2k+1). - Philippe Deléham, Nov 12 2008
O.g.f for column k, k>=0: (1/(1-x)^2)*(x/(1-x))^(2*k). See the G.f. of this array given above by Emeric Deutsch. - Wolfdieter Lang, Jan 18 2013
T(n,k) = (x^(2*k+1))*((1+x)^n-(1-x)^n)/2. - L. Edson Jeffery, Jan 15 2014

More terms from Emeric Deutsch, Apr 01 2005

A110555 Triangle of partial sums of alternating binomial coefficients: T(n, k) = Sum_{j=0..k} binomial(n, j)*(-1)^j, for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0, 1, -8, 28, -56, 70, -56, 28, -8, 1, 0, 1, -9, 36, -84, 126, -126, 84, -36, 9, -1, 0, 1, -10, 45, -120, 210, -252, 210, -120

Offset: 0

Reinhard Zumkeller, Jul 27 2005

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  0;
  [2] 1, -1,  0;
  [3] 1, -2,  1,   0;
  [4] 1, -3,  3,  -1,  0;
  [5] 1, -4,  6,  -4,  1,   0;
  [6] 1, -5, 10, -10,  5,  -1,  0;
  [7] 1, -6, 15, -20, 15,  -6,  1,  0;
  [8] 1, -7, 21, -35, 35, -21,  7, -1,  0.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
Index entries for triangles and arrays related to Pascal's triangle

T(n,1)  = -n + 1 for n>0;
T(n,2)  =  A000217(n-2) for n > 1;
T(n,3)  = -A000292(n-4) for n > 2;
T(n,4)  =  A000332(n-1) for n > 3;
T(n,5)  = -A000389(n-1) for n > 5;
T(n,6)  =  A000579(n-1) for n > 6;
T(n,7)  = -A000580(n-1) for n > 7;
T(n,8)  =  A000581(n-1) for n > 8;
T(n,9)  = -A000582(n-1) for n > 9;
T(n,10) =  A001287(n-1) for n > 10;
T(n,11) = -A001288(n-1) for n > 11;
T(n,12) =  A010965(n-1) for n > 12;
T(n,13) = -A010966(n-1) for n > 13;
T(n,14) =  A010967(n-1) for n > 14;
T(n,15) = -A010968(n-1) for n > 15;
T(n,16) =  A010969(n-1) for n > 16.
Cf. A071919 (variant), A000007 (row sums), A110556 (central terms).
Cf. A008949, A007318.

T := (n, k) -> (-1)^k * binomial(n-1, k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7); # Peter Luschny, Apr 13 2023

T[0, 0] := 1;  T[n_, n_] := 0; T[n_, k_] := (-1)^k*Binomial[n - 1, k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 31 2017 *)

concat(1, for(n=1,10, for(k=0,n, print1(if(k != n, (-1)^k*binomial(n-1,k), 0), ", ")))) \\ G. C. Greubel, Aug 31 2017

T(n, 0) = 1, T(n, n) = 0^n, T(n, k) = -T(n-1, k-1) + T(n-1, k), for 0 < k < n.
T(n, k) = binomial(n-1, k)*(-1)^k, 0 <= k < n, T(n, n) = 0^n.
T(n, n-k-1) = -T(n, k), for 0 < k < n.
T(n, k) = A071919(n, k)*(-1)^k and A071919(n, k) = abs(T(n, k)).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [0, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 05 2005
G.f.: (1 + x*y) / (1 + x*y - x). - R. J. Mathar, Aug 11 2015

Typo in name corrected by Andrey Zabolotskiy, Feb 22 2022

Offset corrected by Peter Luschny, Apr 13 2023

A128908 Riordan array (1, x/(1-x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 10, 6, 1, 0, 5, 20, 21, 8, 1, 0, 6, 35, 56, 36, 10, 1, 0, 7, 56, 126, 120, 55, 12, 1, 0, 8, 84, 252, 330, 220, 78, 14, 1, 0, 9, 120, 462, 792, 715, 364, 105, 16, 1, 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1

Offset: 0

Philippe Deléham, Apr 22 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Row sums give A088305. - Philippe Deléham, Nov 21 2007
Column k is C(n,2k-1) for k > 0. - Philippe Deléham, Jan 20 2012
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
T is the convolution triangle of the positive integers (see A357368). - Peter Luschny, Oct 19 2022

			The triangle T(n,k) begins:
   n\k  0    1    2    3    4    5    6    7    8    9   10
   0:   1
   1:   0    1
   2:   0    2    1
   3:   0    3    4    1
   4:   0    4   10    6    1
   5:   0    5   20   21    8    1
   6:   0    6   35   56   36   10    1
   7:   0    7   56  126  120   55   12    1
   8:   0    8   84  252  330  220   78   14    1
   9:   0    9  120  462  792  715  364  105   16    1
  10:   0   10  165  792 1716 2002 1365  560  136   18    1
  ... reformatted by _Wolfdieter Lang_, Jul 31 2017
From _Peter Luschny_, Mar 06 2022: (Start)
The sequence can also be seen as a square array read by upwards antidiagonals.
   1, 1,   1,    1,    1,     1,     1,      1,      1, ...  A000012
   0, 2,   4,    6,    8,    10,    12,     14,     16, ...  A005843
   0, 3,  10,   21,   36,    55,    78,    105,    136, ...  A014105
   0, 4,  20,   56,  120,   220,   364,    560,    816, ...  A002492
   0, 5,  35,  126,  330,   715,  1365,   2380,   3876, ... (A053126)
   0, 6,  56,  252,  792,  2002,  4368,   8568,  15504, ... (A053127)
   0, 7,  84,  462, 1716,  5005, 12376,  27132,  54264, ... (A053128)
   0, 8, 120,  792, 3432, 11440, 31824,  77520, 170544, ... (A053129)
   0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
    A27,A292, A389, A580,  A582, A1288, A10966, A10968, A165817       (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014.

Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
Cf. A165817 (the main diagonal of the array).

# Computing the rows of the array representation:
S := proc(n,k) option remember;
if n = k then 1 elif k < 0 or k > n then 0 else
S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
for n from 0 to 8 do Arow(n, 9) od; # Peter Luschny, Mar 06 2022
# Uses function PMatrix from A357368.
PMatrix(10, n -> n); # Peter Luschny, Oct 19 2022

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - x)^2/(1 - (2 + y)*x + x^2), {x, 0, nmax}, {y, 0, nmax}], x], y]] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, if(k==0, 0, binomial(n+k-1,2*k-1))), ", "))) \\ G. C. Greubel, Nov 22 2017

from functools import cache
@cache
def A128908(n, k):
    if n == k: return 1
    if (k <= 0 or k > n): return 0
    return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
for n in range(10):
    print([A128908(n, k) for k in range(n+1)]) # Peter Luschny, Mar 07 2022

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A128908 = lambda n,k: T(k,n)
for n in (0..10): print([A128908(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - Philippe Deléham, Oct 19 2008
G.f.: (1-x)^2/(1-(2+y)*x+x^2). - Philippe Deléham, Jan 20 2012
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A001352(n), (-1)^(n+1)*A054888(n+1), (-1)^n*A008574(n), (-1)^n*A084103(n), (-1)^n*A084099(n), A163810(n), A000007(n), A088305(n) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Jan 20 2012
Riordan array (1, x/(1-x)^2). - Philippe Deléham, Jan 20 2012

A221857 Number A(n,k) of shapes of balanced k-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 1, 1, 0, 1, 1, 4, 3, 4, 1, 0, 1, 1, 5, 6, 1, 4, 1, 0, 1, 1, 6, 10, 4, 9, 4, 1, 0, 1, 1, 7, 15, 10, 1, 27, 1, 1, 0, 1, 1, 8, 21, 20, 5, 16, 27, 8, 1, 0, 1, 1, 9, 28, 35, 15, 1, 96, 81, 16, 1, 0, 1, 1, 10, 36, 56, 35, 6, 25, 256, 81, 32, 1, 0

Offset: 0

Alois P. Heinz, Apr 10 2013

			: A(2,2) = 2  : A(2,3) = 3      : A(3,3) = 3          :
:   o     o   :   o    o    o   :   o      o      o   :
:  / \   / \  :  /|\  /|\  /|\  :  /|\    /|\    /|\  :
: o         o : o      o      o : o o    o   o    o o :
:.............:.................:.....................:
: A(3,4) = 6                                          :
:    o        o        o        o       o        o    :
:  /( )\    /( )\    /( )\    /( )\   /( )\    /( )\  :
: o o      o   o    o     o    o o     o   o      o o :
Square array A(n,k) begins:
  1, 1, 1,  1,   1,   1,  1,  1,  1,   1,   1, ...
  1, 1, 1,  1,   1,   1,  1,  1,  1,   1,   1, ...
  0, 1, 2,  3,   4,   5,  6,  7,  8,   9,  10, ...
  0, 1, 1,  3,   6,  10, 15, 21, 28,  36,  45, ...
  0, 1, 4,  1,   4,  10, 20, 35, 56,  84, 120, ...
  0, 1, 4,  9,   1,   5, 15, 35, 70, 126, 210, ...
  0, 1, 4, 27,  16,   1,  6, 21, 56, 126, 252, ...
  0, 1, 1, 27,  96,  25,  1,  7, 28,  84, 210, ...
  0, 1, 8, 81, 256, 250, 36,  1,  8,  36, 120, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Jeffrey Barnett, Counting Balanced Tree Shapes, 2007
Samuele Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv:1107.3472 [math.CO], Apr 2012

Columns k=1-10 give: A000012, A110316, A131889, A131890, A131891, A131892, A131893, A229393, A229394, A229395.
Rows n=0+1, 2-3, give: A000012, A001477, A179865.
Diagonal and upper diagonals give: A028310, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288.
Lower diagonals give: A000012, A000290, A092364(n) for n>1.

A:= proc(n, k) option remember; local m, r; if n<2 or k=1 then 1
      elif k=0 then 0 else r:= iquo(n-1, k, 'm');
      binomial(k, m)*A(r+1, k)^m*A(r, k)^(k-m) fi
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n-1, k]; Binomial[k, m]*a[r+1, k]^m*a[r, k]^(k-m)]]]; Table[a[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 17 2013, translated from Maple *)

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0

Offset: 0

Philippe Deléham, Mar 05 2014

Row sums are powers of 2.

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.

Cf. A095704, A178616.

Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A039948 A triangle related to A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 18, 12, 3, 1, 120, 72, 24, 4, 1, 960, 600, 180, 40, 5, 1, 9360, 5760, 1800, 360, 60, 6, 1, 105840, 65520, 20160, 4200, 630, 84, 7, 1, 1370880, 846720, 262080, 53760, 8400, 1008, 112, 8, 1, 19958400, 12337920, 3810240, 786240, 120960, 15120, 1512, 144, 9, 1

Offset: 0

Wolfdieter Lang

			Triangle begins :
    1;
    1,   1;
    4,   2,   1;
   18,  12,   3,  1;
  120,  72,  24,  4, 1;
  960, 600, 180, 40, 5, 1;
... - _Philippe Deléham_, Nov 08 2011

Seiichi Manyama, Rows n = 0..139, flattened
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.

Cf. A000045, A000142, A005442, A005443, A110313 (row sums).

Diagonals include: A000027, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288, A010965, A010966.

[(Factorial(n)/Factorial(k))*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2022

T[n_,k_]:= (n!/k!)*Fibonacci[n-k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2022 *)

def A039948(n, k): return factorial(n-k)*binomial(n,k)*fibonacci(n-k+1)
flatten([[A039948(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2022

T(n, m) = n!*Fibonacci(n-m+1)/m!, n >= m >= 0.
T(n, 0) = A005442(n).
T(n, 1) = A005443(n).
E.g.f. for column m: x^m/(m!*(1-x-x^2)), m >= 0.
From G. C. Greubel, Nov 20 2022: (Start)
T(n, n-1) = A000027(n).
T(n, n-2) = 4*A000217(n-1), n >= 2.
T(n, n-3) = 18*A000292(n-2), n >= 3.
T(n, n-4) = 5! * A000332(n), n >= 4.
T(n, n-5) = 8 * 5! * A000389(n), n >= 5.
T(n, n-6) = 13 * 6! * A000579(n), n >= 6.
T(n, n-7) = 21 * 7! * A000580(n), n >= 7.
T(n, n-8) = 34 * 8! * A000581(n), n >= 8.
T(n, n-9) = 55 * 9! * A000582(n), n >= 9.
T(n, n-10) = 89 * 10! * A001287(n), n >= 10.
T(n, n-11) = 12 * 12! * A001288(n), n >= 11.
T(n, n-12) = 233 * 12! * A010965(n), n >= 12.
T(n, n-13) = 89 * 13! * A010966(n), n >= 13.
Sum_{k=0..n} T(n, k) = A110313(n). (End)

A101095 Fourth difference of fifth powers (A000584).

Original entry on oeis.org

1, 28, 121, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.

The (Worpitzky/Euler/Pascal Cube) "MagicNKZ" algorithm is: MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n, with k>=0, n>=1, z>=0. MagicNKZ is used to generate the n-th accumulation sequence of the z-th row of the Euler Triangle (A008292). For example, MagicNKZ(3,k,0) is the 3rd row of the Euler Triangle (followed by zeros) and MagicNKZ(10,k,1) is the partial sums of the 10th row of the Euler Triangle. This sequence is MagicNKZ(5,k-1,2).

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive Machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Fourth differences of A000584, third differences of A022521, second differences of A101098, and first differences of A101096.
For other sequences based upon MagicNKZ(n,k,z):
...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8
--------------------------------------------------------------------------------------
z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
z =  2 | A000027 | A005408 | A008458 | A101103 | thisSeq | ....... | ....... | .......
z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | .......
z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | .......
z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | .......
z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182
z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178
z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524
z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016
z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542
z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636
z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642
z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647
z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......
--------------------------------------------------------------------------------------
Cf. A047969.

I:=[1,28,121,240,360]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, May 07 2015

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 2, 2}, {k, 0, 34}]
CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)
Join[{1,28,121,240},Differences[Range[50]^5,4]] (* or *) LinearRecurrence[{2,-1},{1,28,121,240,360},50] (* Harvey P. Dale, Jun 11 2016 *)

a(n)=if(n>3, 120*n-240, 33*n^2-72*n+40) \\ Charles R Greathouse IV, Oct 11 2015

[1,28,121]+[120*(k-2) for k in range(4,36)] # Danny Rorabaugh, Apr 23 2015

a(k+1) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n; n = 5, z = 2.
For k>3, a(k) = Sum_{j=0..4} (-1)^j*binomial(4, j)*(k - j)^5 = 120*(k - 2).
a(n) = 2*a(n-1) - a(n-2), n>5. G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (1-x)^2. - Colin Barker, Mar 01 2012

MagicNKZ material edited, Crossrefs table added, SeriesAtLevelR material removed by Danny Rorabaugh, Apr 23 2015

Name changed and keyword 'uned' removed by Danny Rorabaugh, May 06 2015

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A135278 Triangle read by rows, giving the numbers T(n,m) = binomial(n+1, m+1); or, Pascal's triangle A007318 with its left-hand edge removed.

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A078812 Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

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A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

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A034867 Triangle of odd-numbered terms in rows of Pascal's triangle.

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A110555 Triangle of partial sums of alternating binomial coefficients: T(n, k) = Sum_{j=0..k} binomial(n, j)*(-1)^j, for n >= 0, 0 <= k <= n.

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