A091894 -id:A091894 - cp's OEIS frontend

A007564 Shifts left when INVERT transform applied thrice.

1, 1, 4, 19, 100, 562, 3304, 20071, 124996, 793774, 5120632, 33463102, 221060008, 1473830308, 9904186192, 67015401391, 456192667396, 3122028222934, 21467769499864, 148246598341018, 1027656663676600, 7148588698592956, 49884553176689584

Offset: 0

N. J. A. Sloane

More generally, coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+2)*x+1))/(2*m*x) are given by a(n) = Sum_{k=0..n} (m+1)^k*N(n,k) where N(n,k) = (1/n)*binomial(n,k)*binomial(n,k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 24 2003
If y = x*A(x) then 3*y^2 - (1+2*x)*y + x = 0 and x = y*(1-3*y)/(1-2*y). - Michael Somos, Sep 28 2003
The sequence 0,1,4,19,... with g.f. (1-4*x-sqrt(1-8*x+4*x^2))/(6*x) and has a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k)*C(k)*4^(n-1-2*k)*3^k. a(n+1) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*C(k)*4^(n-2*k)*3^k counts Motzkin paths of length n in which the level steps have 4 colors and the up steps have 3. It is the binomial transform of A107264 and corresponds to the series reversion of x/(1+4*x+3*x^2). - Paul Barry, May 18 2005
The Hankel transform of this sequence is 3^binomial(n+1,2). - Philippe Deléham, Oct 29 2007
a(n) is the number of Schroder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 2 colors. Example: a(2)=4 because we have UDUD, UUDD, UBD, and URD, where U=(1,1), D=(1,-1), while B (R) is a blue (red) (2,0)-step. - Emeric Deutsch, May 02 2011
a(n) is the number of Schroder paths of semilength n-1 in which the (2,0)-steps at level 0 come in 3 colors and those at a higher level come in 2 colors. Example: a(3)=19 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 3^2 = 9 paths of shape HH, 3 paths of shape HUD, 3 paths of shape UDH, 2 paths of shape UHD, and 1 path of each of the shapes UDUD and UUDD. - Emeric Deutsch, May 02 2011
From David Callan, Jun 21 2013: (Start)
a(n) = number of (left) planted binary trees with n edges in which each vertex has a designated favorite neighbor. Planted binary trees are counted by the Catalan numbers A000108.
Example: for n=2, there are 2 planted binary trees: edges LL and LR from the root (L=left, R=right). Each has just one vertex with 2 neighbors, and so a(2)=4.
Proof outline: each vertex has 1,2 or 3 neighbors. Let X (resp. Y) denote the number of vertices with 2 (resp. 3) neighbors. Then X + 2Y = n - 1 (split the non-root edges into pairs with a common parent vertex and singletons). Thus the number of choices for designating favorite neighbors is 2^X * 3^Y = 2^(n-1)(3/4)^Y. The distribution for Y is known because, under the rotation correspondence, a.k.a. the deBruijn-Morselt bijection, vertices with 2 children in an n-edge planted binary tree correspond to DDUs in a Dyck path, and DDUs have the Touchard distribution (A091894) with gf F(x,y) = (1-2x+2xy - sqrt(1-4x+4x^2-4x^2 y))/(2xy). The desired g.f., Sum_{n>=1} a(n)*x^n, is therefore 1/2*(F(2x,3/4)-1). (End)

			G.f. = 1 + x + 4*x^2 + 19*x^3 + 100*x^4 + 562*x^5 + 3304*x^6 + 20071*x^7 + 124996*x^8 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 to 100 by T. D. Noe)
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.8.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 arXiv:1608.02448 [math.CO], 2016. Eq. (1.13) a=1, b=3.
C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 443
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 11.
N. J. A. Sloane, Transforms

Cf. A047891, A054872.

A007564_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+3*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A007564_list(21); # Peter Luschny, May 19 2011

a[0]=1; a[1]=1; a[n_]/;n>=2 := a[n] = a[n-1] + 3 Sum[a[k-1]a[n-k],{k,n-1}] ; Table[a[n],{n,0,10}] (* David Callan, Aug 25 2009 *)
Table[Hypergeometric2F1[-n, 1 - n, 2, 3], {n, 0, 22}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[(2^n (LegendreP[n+1, 2] - LegendreP[n-1, 2]) + 2 KroneckerDelta[n])/(6n+3), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
CoefficientList[Series[(1+2x-Sqrt[1-8x+4x^2])/(6x),{x,0,30}],x] (* Harvey P. Dale, Feb 07 2016 *)

{a(n) = if( n<1, n==0, sum( k=0, n, 3^k * binomial( n, k) * binomial( n, k+1)) / n)} /* Michael Somos, Sep 28 2003 */

{a(n) = if( n<0, 0, n++; polcoeff( serreverse( x * (1 - 3*x) / (1 - 2*x) + x * O(x^n)), n))} /* Michael Somos, Sep 28 2003 */

a(n) = (2^n*(pollegendre(n+1,2)-pollegendre(n-1,2)) + 2*(n==0))/(6*n+3); \\ Michel Marcus, Nov 02 2015

x='x+O('x^100); Vec((1+2*x-sqrt(1-8*x+4*x^2))/(6*x)) \\ Altug Alkan, Nov 02 2015

G.f.: (1+2*x-sqrt(1-8*x+4*x^2))/(6*x). - Emeric Deutsch, Nov 03 2001
a(0)=1; for n>=1, a(n) = Sum_{k=0..n} 3^k*N(n,k) where N(n,k) = (1/n)*binomial(n, k)*binomial(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 24 2003
a(n) = Sum_{k=0..n} A088617(n, k)*3^k*(-2)^(n-k). - Philippe Deléham, Jan 21 2004
With offset 1: a(1) = 1, a(n) = -2*a(n-1) + 3*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
D-finite with recurrence a(n) = (4*(2n-1)*a(n-1) - 4*(n-2)*a(n-2)) / (n+1) for n>=2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005
From Paul Barry, Dec 15 2008: (Start)
G.f.: 1/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x........ (continued fraction).
The g.f. of a(n+1) is 1/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2.... (continued fraction). (End)
a(0) = 1, for n>=1, 3a(n) = A047891(n). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
a(n) = upper left term in M^n, M = the production matrix:
1, 1
3, 3, 3
1, 1, 1, 1
3, 3, 3, 3, 3
1, 1, 1, 1, 1, 1
...
- Gary W. Adamson, Jul 08 2011
G.f.: A(x)= (1+2*x-sqrt(1-8*x+4*x^2))/(6*x)= 1/G(0); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
a(n) ~ sqrt(6+4*sqrt(3))*(4+2*sqrt(3))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012
a(n) = 2^n/sqrt(3)*LegendreP(n,-1,2) for n >= 1, where LegendreP is the associated Legendre function of the first kind, in Maple's notation. - Robert Israel, Mar 24 2015

A133156 Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials of the second kind with exponents in decreasing order.

Original entry on oeis.org

1, 2, 4, -1, 8, -4, 16, -12, 1, 32, -32, 6, 64, -80, 24, -1, 128, -192, 80, -8, 256, -448, 240, -40, 1, 512, -1024, 672, -160, 10, 1024, -2304, 1792, -560, 60, -1, 2048, -5120, 4608, -1792, 280, -12, 4096, -11264, 11520, -5376, 1120, -84, 1

Offset: 0

Gary W. Adamson, Dec 16 2007

The Chebyshev polynomials of the second kind are defined by the recurrence relation: U(0,x) = 1; U(1,x) = 2x; U(n+1,x) = 2x*U(n,x) - U(n-1,x).
From Gary W. Adamson, Nov 28 2008: (Start)
Triangle read by rows, unsigned = A000012 * A028297.
Row sums of absolute values give the Pell series, A000129.
(End)
The row sums are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...}.
Triangle, with zeros omitted, given by (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 27 2011
Coefficients in the expansion of sin((n+1)*x)/sin(x) in descending powers of cos(x). The length of the n-th row is A008619(n). - Jianing Song, Nov 02 2018

			The first few Chebyshev polynomials of the second kind are
    1;
    2x;
    4x^2 -    1;
    8x^3 -    4x;
   16x^4 -   12x^2 +   1;
   32x^5 -   32x^3 +   6x;
   64x^6 -   80x^4 +  24x^2 -   1;
  128x^7 -  192x^5 +  80x^3 -   8x;
  256x^8 -  448x^6 + 240x^4 -  40x^2 +  1;
  512x^9 - 1024x^7 + 672x^5 - 160x^3 + 10x;
  ...
From _Roger L. Bagula_ and _Gary W. Adamson_: (Start)
     1;
     2;
     4,    -1;
     8,    -4;
    16,   -12,    1;
    32,   -32,    6;
    64,   -80,   24,   -1;
   128,  -192,   80,   -8;
   256,  -448,  240,  -40,  1;
   512, -1024,  672, -160, 10;
  1024, -2304, 1792, -560, 60, -1; (End)
From  _Philippe Deléham_, Dec 27 2011: (Start)
Triangle (2, 0, 0, 0, 0, ...) DELTA (0, -1/2, 1/2, 0, 0, 0, 0, 0, ...) begins:
   1;
   2,   0;
   4,  -1,  0;
   8,  -4,  0,  0;
  16, -12,  1,  0,  0;
  32, -32,  6,  0,  0,  0;
  64, -80, 24, -1,  0,  0,  0; (End)

Tracale Austin, Hans Bantilan, Isao Jonas and Paul Kory, The Pfaffian Transformation, Journal of Integer Sequences, Vol. 12 (2009), page 25
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014. - From _Tom Copeland_, Oct 11 2014
Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120-135. MR3149030
Caglar Koca and Ozgur B. Akan, Modelling 1D Partially Absorbing Boundaries for Brownian Molecular Communication Channels, arXiv:2402.15888 [q-bio.MN], 2024. See p. 9.
Wikipedia, Chebyshev polynomials

Cf. A038207, A053117.
Cf. A018297, A000129. - Gary W. Adamson, Nov 28 2008
Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851. - Philippe Deléham, Sep 12 2009
Cf. A091894, A097610, A099089, A207538.

t[n_, m_] = (-1)^m*Binomial[n - m, m]*2^(n - 2*m);
Table[Table[t[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula, Dec 19 2008 *)

A generating function for U(n) is 1/(1 - 2tx + t^2). Given A038207, shift down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row, then insert alternate signs.
T(n,m) = (-1)^m*binomial(n - m, m)*2^(n - 2*m). - Roger L. Bagula and Gary W. Adamson, Dec 19 2008
From Tom Copeland, Feb 11 2016: (Start)
Shifted o.g.f.: G(x,t) = x/(1 - 2x + tx^2).
A053117 is a reflected, aerated version of this entry; A207538, an unsigned version; and A099089, a reflected, shifted version.
The compositional inverse of G(x,t) is Ginv(x,t) = ((1 + 2x) - sqrt((1 + 2x)^2 - 4tx^2))/(2tx) = x - 2x^2 + (4 + t)x^3 - (8 + 6t)x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0.). Cf. A097610 with h_1 = -2 and h_2 = t. (End)

More terms from Philippe Deléham, Sep 12 2009

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

Original entry on oeis.org

1, 2, 4, 1, 8, 4, 16, 12, 1, 32, 32, 6, 64, 80, 24, 1, 128, 192, 80, 8, 256, 448, 240, 40, 1, 512, 1024, 672, 160, 10, 1024, 2304, 1792, 560, 60, 1, 2048, 5120, 4608, 1792, 280, 12, 4096, 11264, 11520, 5376, 1120, 84, 1, 8192, 24576, 28160, 15360

Offset: 1

Clark Kimberling, Feb 18 2012

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is the triangle given by (2, 0, 0, 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, Mar 04 2012
The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and  along (first layer) skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. - Zagros Lalo, Jul 31 2018
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.414213562373095... (A014176: Decimal expansion of the silver mean, 1+sqrt(2)), when n approaches infinity. - Zagros Lalo, Jul 31 2018

			First seven rows:
1
2
4...1
8...4
16..12..1
32..32..6
64..80..24..1
(2, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, ...) begins:
    1
    2,   0
    4,   1,  0
    8,   4,  0, 0
   16,  12,  1, 0, 0
   32,  32,  6, 0, 0, 0
   64,  80, 24, 1, 0, 0, 0
  128, 192, 80, 8, 0, 0, 0, 0

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358.

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
S. Halici, On some Pell polynomials , Acta Universitatis Apulensis, No. 29/2012, pp. 105-112.
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n

Cf. A028297, A207537, A133156, A038207, A099089.
Cf. A053117, A097610, A091894.
Cf. A013609, A038207.
Cf. A128099.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)
t[n_, k_] := t[n, k] = 2^(n - 2 k) * (n -  k)!/((n - 2 k)! k!) ; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]} ]  // Flatten (* Zagros Lalo, Jul 31 2018 *)

u(n,x) = u(n-1,x)+(x+1)*v(n-1,x), v(n,x) = u(n-1,x)+v(n-1,x), where u(1,x) = 1, v(1,x) = 1. Also, A207538 = |A133156|.
From Philippe Deléham, Mar 04 2012: (Start)
With 0<=k<=n:
Mirror image of triangle in A099089.
Skew version of A038207.
Riordan array (1/(1-2*x), x^2/(1-2*x)).
G.f.: 1/(1-2*x-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190958(n+1), A127357(n), A090591(n), A089181(n+1), A088139(n+1), A045873(n+1), A088138(n+1), A088137(n+1), A099087(n), A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively.
T(n,k) = 2*T(n-1,k) + T(-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n, k) = 0 if k<0 or if k>n. (End)
T(n,k) = A013609(n-k, n-2*k+1). - Johannes W. Meijer, Sep 05 2013
From Tom Copeland, Feb 11 2016: (Start)
A053117 is a reflected, aerated and signed version of this entry. This entry belongs to a family discussed in A097610 with parameters h1 = -2 and h2 = -y.
Shifted o.g.f.: G(x,t) = x / (1 - 2 x - t x^2).
The compositional inverse of G(x,t) is Ginv(x,t) = -[(1 + 2x) - sqrt[(1+2x)^2 + 4t x^2]] / (2tx) = x - 2 x^2 + (4-t) x^3 - (8-6t) x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0).
(End)

A051288 Triangle read by rows: T(n,k) = number of paths of n upsteps U and n downsteps D that contain k UUDs.

Original entry on oeis.org

1, 2, 4, 2, 8, 12, 16, 48, 6, 32, 160, 60, 64, 480, 360, 20, 128, 1344, 1680, 280, 256, 3584, 6720, 2240, 70, 512, 9216, 24192, 13440, 1260, 1024, 23040, 80640, 67200, 12600, 252, 2048, 56320, 253440, 295680, 92400, 5544, 4096, 135168, 760320

Offset: 0

Clark Kimberling

By reading paths backward, the UUD in the name could be replaced by DDU.
Or, triangular array T read by rows: T(n,k)=P(2n,n,4k), where P(n,k,c)=number of vectors (x(1),x(2,),...,x(n)) of k 1's and n-k 0's such that x(i)=x(n+1-i) for exactly c values of i. P(n,k,n) counts palindromes.
In nuclear magnetic resonance of n coupled spin-1/2 nuclides, T(n,k) is the number of zero-quantum transitions with combination index k. See the [Sykora (2007)] link, containing also yet another interpretation in terms of pairs of binary n-tuples. - Stanislav Sykora, Apr 27 2012
Let u - (u_1, u_2, u_3, ..., u_{2n}) be a binary vector containing n 0's and n 1's. Define a mismatch to be an adjacent pair (u_{2i-1}, u_{2i}) which is neither 0,1 nor 1,0 (think "socks"). Then T(n,k) = number of u's with k mismatches. - N. J. A. Sloane, Nov 03 2017 following an email from Bill Gosper
From Colin Defant, Sep 16 2018: (Start)
Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n] with k valleys such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n] with k valleys such that s(pi) avoids the patterns 132, 312, and 321.
T(n,k) is the number of permutations of [n] with k valleys that avoid the patterns 1342, 3142, 3412, and 3421. (End)
T(n,k) is the number of card sequences for a balanced deck of 2n cards that end up with k pairs of cards in the black pile at the end of Stewart James' classic Miraskill card trick. See Table 2 in the [Tuenter (2024)] link. - Hans J. H. Tuenter, Dec 27 2024

			Table begins
n | k=0    1    2    3
--+-------------------
0 |   1
1 |   2
2 |   4    2
3 |   8   12
4 |  16   48    6
5 |  32  160   60
6 |  64  480  360   20
7 | 128 1344 1680  280
...
a(2,1)=2 because UUDD, DUUD each have one UUD.

Stanislav Sykora, Table of n, a(n) for n = 0..2600
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.
S. Elizalde, E. Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7 eq (10)
Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2.
Stanislav Sykora, Triangle T(n,k) for rows n = 0..100
Stanislav Sykora, p-Quantum Transitions and a Combinatorial Identity, Stan's Library II, 2007, Identity (1) for p=0.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Hans J. H. Tuenter, Combinatorial Analysis of a Classic Card Trick, Mathematics Magazine, 97(5):551-558, December 2024.

Row sums are the (even) central binomial coefficients A000984. A091894 gives the distribution of the parameter "number of DDUs" on Dyck paths.

Table[Binomial[n, 2k]2^(n-2k)Binomial[2k, k], {n, 0, 15}, {k, 0, n/2}]

T(n, k) = binomial(n, 2*k)*2^(n-2*k)*binomial(2*k, k).

G.f.: (1-4*x+4*x^2*(1-y))^(-1/2) = Sum_{n>=0, k>=0} a(n, k)*x^n*y^k.

Additional comments from David Callan, Aug 28 2004

A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 8, 17, 12, 4, 1, 16, 46, 44, 20, 5, 1, 32, 120, 150, 90, 30, 6, 1, 64, 304, 482, 370, 160, 42, 7, 1, 128, 752, 1476, 1412, 770, 259, 56, 8, 1, 256, 1824, 4344, 5068, 3402, 1428, 392, 72, 9, 1, 512, 4352, 12368, 17285, 14000, 7168, 2436

Offset: 1

R. J. Mathar, Feb 21 2010

From Johannes W. Meijer, May 06 2011: (Start)
The Row1, Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Kn3, Kn4 and Ca1 triangle sums link A175136 with several sequences, see the crossrefs. For the definitions of these triangle sums see A180662.
It is remarkable that the coefficients of the right hand columns of A175136, and subsequently those of triangle A175136, can be generated with the aid of the row coefficients of A091894. For the fourth, fifth and sixth right hand columns see A162148, A190048 and A190049. The a(n) formulas of the right hand columns lead to an explicit formula for the T(n,k), see the formulas and the second Maple program. (End)
Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2011.
T(n,k) is the number of noncrossing partitions of n containing k runs, where a block forms a run if it consists of an interval of integers. For example, T(4,2)=6 counts 1/234, 12/34, 123/4, 1/24/3, 13/2/4, 14/2/3. - David Callan, Oct 14 2012

			Triangle starts
    1;
    1,    1;
    2,    2,    1;
    4,    6,    3,    1;
    8,   17,   12,    4,    1;
   16,   46,   44,   20,    5,    1;
   32,  120,  150,   90,   30,    6,    1;
   64,  304,  482,  370,  160,   42,    7,    1;
  128,  752, 1476, 1412,  770,  259,   56,    8,    1;
Triangle (0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  4,  6,  3,  1;
  0,  8, 17, 12,  4,  1; ... - _Philippe Deléham_, Oct 29 2011

David Callan, A bijection on Dyck paths and its cycle structure, El. J. Combinat. 14 (2007) # R28.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
David Callan and Emeric Deutsch, The Run Transform, arXiv:1112.3639 [math.CO], 2011.
K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Nonleft peaks in Dyck paths: a combinatorial approach, Discrete Math., 337 (2014), 97-105.

Triangle columns: A000012, A000027, A002378, A162148, A190048, A190049, A011782, A190050, A190051.
Triangle sums (see the comments): A000108 (Row1), A005043 (Related to Kn11, Kn12, Kn13 and Kn4), A007477 (Related to Kn21, Kn22, Kn23 and Kn3), A099251 (Kn4), A166300 (Ca1). - Johannes W. Meijer, May 06 2011
Cf. A000108 (row sums), A196182

lco := proc(siz,leav) (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/2/x ; coeftayl(%,x=0,siz ) ; coeftayl(%,y=0,leav ) ; end proc: seq(seq(lco(n,k),k=1..n),n=1..9) ;
T := proc(n, k): add(A091894(n-k, k1)*binomial(n-k1-1, n-k), k1=0..floor((n-k)/2)) end: A091894 := proc(n, k): if n=0 and k=0 then 1 elif n=0 then 0 else 2^(n-2*k-1)* binomial(n-1, 2*k) * binomial(2*k, k)/(k+1) fi end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, May 06 2011, revised Nov 23 2012

A091894[n_, k_] := 2^(n - 2*k - 1)*Binomial[n - 1, 2*k]*(Binomial[2*k, k]/(k + 1)); t[n_, k_] := Sum[A091894[n - k, k1]*Binomial [n - k1 - 1, n - k], {k1, 0, (n - k)/2}]; t[n_, n_] = 1; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten(* Jean-François Alcover, Jun 13 2013, after Johannes W. Meijer *)

G.f.: (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/(2*x).

T(n,k) = Sum_{k1=0..floor((n-k)/2)} A091894(n-k, k1)*binomial(n-k1-1, n-k), 1 <= k <= n. - Johannes W. Meijer, May 06 2011

Variable names changed by Johannes W. Meijer, May 06 2011

A068763 Irregular triangle of the Fibonacci polynomials of A011973 multiplied diagonally by the Catalan numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 1, 14, 20, 6, 42, 70, 30, 2, 132, 252, 140, 20, 429, 924, 630, 140, 5, 1430, 3432, 2772, 840, 70, 4862, 12870, 12012, 4620, 630, 14, 16796, 48620, 51480, 24024, 4620, 252, 58786, 184756, 218790

Offset: 0

Wolfdieter Lang, Mar 04 2002

The row length sequence of this array is [1,2,2,3,3,4,4,5,5,...] = A008619(n+2), n>=0.
The row polynomials p(n,x) := Sum_{m=0..floor((n+1)/2)} a(n,m)*x^m produce, for x = (b-a^2)/a^2 (not 0), the two parameter family of sequences K(a,b; n) := (a^(n+1))*p(n,(b-a^2)/a^2) with g.f. K(a,b; x) := (1-sqrt(1-4*x*(a+x*(b-a^2))))/(2*x).
Some members are: K(1,1; n)=A000108(n) (Catalan), K(1,2; n)=A025227(n-1), K(2,1; n)=A025228(n-1), K(1,3; n)=A025229(n-1), K(3,1; n)=A025230(n-1). For a=b=2..10 the sequences K(a,a; n)/a are A068764-A068772.
The column sequences (without leading 0's) are: A000108 (Catalan), A000984 (central binomial), A002457, 2*A002802, 5*A020918, 14*A020920, 42*A020922, ...
a(n,m) is the number of ways to designate exactly m cherries over all binary trees with n internal nodes.  A cherry is an internal node whose descendants are both external nodes.  Cf. A091894 which gives the number of binary trees with m cherries. - Geoffrey Critzer, Jul 24 2020
This irregular triangle is essentially that of A011973 with its diagonals multiplied by the Catalan numbers of A000108. The diagonals of this triangle are then rows of the Pascal matrix A007318 multiplied by the Catalan numbers. - Tom Copeland, Dec 23 2023

			The irregular triangle begins:
   n\m    0     1     2     3    4   5
   0:     1
   1:     1     1
   2:     2     2
   3:     5     6     1
   4:    14    20     6
   5:    42    70    30     2
   6:   132   252   140    20
   7:   429   924   630   140    5
   8:  1430  3432  2772   840   70
   9:  4862 12870 12012  4620  630  14
  10: 16796 48620 51480 24024 4620 252
  ...
p(3,x) = 5 + 6*x + x^2.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 213.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A025227(n-1) (row sums).
Cf. A000007(n) (alternating row sums).
Cf. A000108, A001263, A007318, A011973, A033832, A034867, A133437 and A134264.

nn = 10; b[z_] := (1 - Sqrt[1 - 4 z])/(2 z);Map[Select[#, # > 0 &] &,
CoefficientList[Series[v b[v z] /. v -> (1 + u z ), {z, 0, nn}], {z, u}]] // Grid (* Geoffrey Critzer, Jul 24 2020 *)

a(n, m) = binomial(n+1-m, m)*C(n-m) if 0 <= m <= floor((n+1)/2), otherwise 0, with C(n) := A000108(n) (Catalan).
G.f. for column m=1, 2, ...: (x^(2*m-1))*C(m-1)/(1-4*x)^((2*m-1)/2); m=0: c(x), g.f. for A000108 (Catalan).
G.f. for row polynomials p(n, x): c(z) + x*z*c(x*(z^2)/(1-4*z))/sqrt(1-4*z) = (1-sqrt(1-4*z*(1+x*z)))/(2*z), where c(x) is the g.f. of A000108 (Catalan).
G.f. for triangle: (1 - sqrt(1 - 4*x (1 + y*x)))/(2*x). - Geoffrey Critzer, Jul 24 2020
The alternating row sums are {1,repeat 0} = {A000007(n)}{n>=0}. From the second comment p(n, x=-1) = K(1,0; n), with g.f. K(1,0; x) = 1. Or from the g.f. of the triangle with y = -1. Thanks to Aaron Li for asking for a proof. - _Wolfdieter Lang, Jan 21 2023
The series expansion of f(x) = (1 + 2sx - sqrt(1 + 4sx + 4d^2x^2))/(2x) at x = 0 is (s^2 - d^2) x + (2 d^2s - 2 s^3) x^2 + (d^4 - 6 d^2 s^2 + 5 s^4) x^3  + (-6 d^4 s + 20 d^2 s^3 - 14 s^5) x^4 + ..., containing the coefficients of this array. With s = (a+b)/2 and d = (a-b)/2, then f(x)/ab = g(x) = (1 + (a+b)x - sqrt((1+(a+b)x)^2 - 4abx^2))/(2abx) = x - (a + b) x^2 + (a^2 + 3 a b + b^2) x^3 - (a^3 + 6 a^2 b + 6 a b^2 + b^3) x^4 + ..., containing the Narayana polynomials of A001263, which can be simply transformed into A033282. The compositional inverse about the origin of g(x) is g^(-1)(x) = x/((1-ax)(1-bx)) = x/((1-(s+d)x)(1-(s-d)x)) = x + (a + b) x^2 + (a^2 + a b + b^2) x^3 + (a^3 + a^2 b + a b^2 + b^3) x^4 + ..., containing the complete homogeneous symmetric polynomials h_n(a,b) = (a^n - b^n)/(a-b), which are the polynomials of A034867 when expressed in s and d, e.g., ((s + d)^7 - (s - d)^7)/(2 d) = d^6 + 21 d^4 s^2 + 35 d^2 s^4 + 7 s^6. A133437 and A134264 for compositional inversion of o.g.f.s can be used to relate the sets of polynomials above. - Tom Copeland, Nov 28 2023

Title changed by Tom Copeland, Dec 23 2023

A162148 a(n) = n(n+1)(5*n+7)/6.

Original entry on oeis.org

0, 4, 17, 44, 90, 160, 259, 392, 564, 780, 1045, 1364, 1742, 2184, 2695, 3280, 3944, 4692, 5529, 6460, 7490, 8624, 9867, 11224, 12700, 14300, 16029, 17892, 19894, 22040, 24335, 26784, 29392, 32164, 35105, 38220, 41514, 44992, 48659, 52520, 56580

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Partial sums of A147875.
Equals the fourth right hand column of A175136 for n>=1. - Johannes W. Meijer, May 06 2011
a(n) is the number of triples (w,x,y) havingt all terms in {0,...,n} and x+y>w. - Clark Kimberling, Jun 14 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002412, A002413, A006331, A016061, A033994.
Cf. A162147, A175136, A190048, A190049, A254407.
Related to A000292 and A091894.

[n*(n+1)*(5*n+7)/6: n in [0..50]]; // Vincenzo Librandi, May 07 2011

A162148:= n-> n*(n+1)*(5*n+7)/6; seq(A162148(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[(n(n+1)(5n+7))/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,4,17,44}, 50] (* Harvey P. Dale, May 20 2014 *)

a(n)=n*(n+1)*(5*n+7)/6 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(5*n+7)/6 for n in (0..50)] # G. C. Greubel, Mar 31 2021

a(n) = A162147(n) + A000217(n).
From Johannes W. Meijer, May 06 2011: (Start)
G.f.: x*(4+x)/(1-x)^4.
a(n) = 4*binomial(n+2,3) + binomial(n+1,3).
a(n) = A091894(3,0)*binomial(n+2,3) + A091894(3,1)*binomial(n+1,3). (End)
a(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i).
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4), a(0)=0, a(1)=4, a(2)=17, a(3)=44. - Harvey P. Dale, May 20 2014
E.g.f.: x*(24 +27*x +5*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

Definition rephrased by R. J. Mathar, Jun 27 2009

A108838 Triangle of Dyck paths counted by number of long interior inclines.

Original entry on oeis.org

2, 3, 2, 4, 8, 2, 5, 20, 15, 2, 6, 40, 60, 24, 2, 7, 70, 175, 140, 35, 2, 8, 112, 420, 560, 280, 48, 2, 9, 168, 882, 1764, 1470, 504, 63, 2, 10, 240, 1680, 4704, 5880, 3360, 840, 80, 2, 11, 330, 2970, 11088, 19404, 16632, 6930, 1320, 99, 2

Offset: 2

David Callan, Jul 25 2005

T(n,k) is the number of Dyck n-paths (A000108) containing k long interior inclines. An incline is an ascent or a descent where an ascent (resp. descent) is a maximal sequence of contiguous upsteps (resp. downsteps). An incline is long if it consists of at least 2 steps and is interior if it does not start or end the path.
T(n,k) is the number of Dyck (n+1)-paths whose last descent has length 2 and which contain n-k peaks. For example T(3,0)=3 counts UUDUDUDD, UDUUDUDD, UDUDUUDD. - David Callan, Jul 03 2006
T(n,k) is the number of parallelogram polyominoes of semiperimeter n+1 having k corners. - Emeric Deutsch, Oct 09 2008
T(n,k) is the number of rooted ordered trees with n non-root nodes and k leaves; see example. - Joerg Arndt, Aug 18 2014

			Table begins
\ k..0....1....2....3....4....5
n\
2 |..2
3 |..3....2
4 |..4....8....2
5 |..5...20...15....2
6 |..6...40...60...24....2
7 |..7...70..175..140...35....2
The paths UUUDDD, UUDUDD, UDUDUD have no long interior inclines; so T(3,0)=3.
From _Joerg Arndt_, Aug 18 2014: (Start)
The rooted ordered trees with n=3 nodes, as (preorder-) level sequences, together with their number of leaves, and an ASCII rendering, are:
:
:     1:  [ 0 1 1 1 ]   2
:  O--o
:  .--o
:  .--o
:
:     2:  [ 0 1 1 2 ]   2
:  O--o
:  .--o--o
:
:     3:  [ 0 1 2 1 ]   1
:  O--o--o
:  .--o
:
:     4:  [ 0 1 2 2 ]   1
:  O--o--o
:     .--o
:
:     5:  [ 0 1 2 3 ]   1
:  O--o--o--o
:
This gives [3, 2], row n=3 of the triangle.
(End)

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150, flattened)
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Tewodros Amdeberhan, Victor H. Moll, and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, arXiv:1202.1203 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 19 2012
Tewodros Amdeberhan, Victor H. Moll, and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, Online Journal of Analytic Combinatorics, Issue 8, 2013.
David Callan, Some Identities for the Catalan and Fine Numbers, arXiv:math/0502532 [math.CO], 2005.
M. Delest, J. P. Dubernard, and I. Dutour, Parallelogram polyominoes and corners, J. Symbolic Computation, 20(1995),503-515. [From _Emeric Deutsch_, Oct 09 2008]
M. P. Delest, D. Gouyou-Beauchamps, and B. Vauquelin, Enumeration of parallelogram polyominoes with given bond and site parameter, Graphs and Combinatorics, 3 (1987), 325-339.
Emeric Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
T. Doslic, Handshakes across a (round) table, JIS 13 (2010) #10.2.7.
Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, Counting pattern-avoiding permutations by big descents, arXiv:2408.15111 [math.CO], 2024. See pp. 6, 18, 27.

Row sums are the Catalan numbers A000108. Column k=1 is A007290, k=2 is A006470. The Narayana numbers A001263 count Dyck paths by number of long nonterminal inclines. A091894 (Touchard distribution) counts Dyck paths by number of long nonterminal descents.

Cf. A145596.

T:=(n,k)->2*binomial(n-1,k)*binomial(n,k+2)/(n-1): for n from 2 to 11 do seq(T(n,k),k=0..n-2) od; # yields sequence in triangular form; Emeric Deutsch, Jul 23 2006

T[n_, 0] = n;
T[n_, k_] := T[n, k] = If[k == n-2, 2, T[n, k-1](n-k-1)(n-k)/(k(k+2))];
Table[T[n, k], {n, 2, 11}, {k, 0, n-2}] // Flatten (* Jean-François Alcover, Jul 27 2018, after Werner Schulte *)

T(n, k) = 2*binomial(n+1, k+2)*binomial(n-2, k)/(n+1).
G.f.: G(z, t) = Sum_{n>=1, k>=0} T(n, k)*z^n*t^k satisfies z - ( (1-z)^2 - (2*t-t^2)*z^2 )*G + (t^2*z)*G^2 = 0.
G.f.: 1+z(1+r)^2, where r=r(t,z) is the Narayana function defined by (1+r)*(1+tr)z=r, r(t,0)=0. - Emeric Deutsch, Jul 23 2006
For n >= 0, the row polynomials Sum_{k=0..n} T(n+2,k)*x^k = (2/(n+1))*(1-x)^n*P(n,2,1,(1+x)/(1-x)), where P(n,a,b,x) denotes the Jacobi polynomial. It follows that the row polynomials have negative real zeros. - Peter Bala, Jan 21 2008
The trivariate g.f. G=G(t,s,z) of Dyck paths with respect to number of DUU's (marked by t), number of DDU's (marked by s) and semilength (marked by z) satisfies G = 1 + z*G + z^2*(1+t*(G-1))*(1+s*(G-1))/(1-z*(1+t*s*(G-1))) (the number of long interior inclines is equal to the number of DUU and DDU's). - Emeric Deutsch, Oct 09 2008
Recurrence: T(n, k) = T(n, k-1)*(n-1-k)*(n-k)/(k*(k+2)) for k > 0 and n >= 2. - Werner Schulte, Jan 04 2017
The array can be extended to negative values of n: T(-n,k) = 2*binomial(-n+1, k+2)*binomial(-n-2, k)/(-n+1) = -A145596(n+k,k+1) for n >= 2. - Peter Bala, Apr 26 2022

A190048 Expansion of (8+6*x)/(1-x)^5.

Original entry on oeis.org

8, 46, 150, 370, 770, 1428, 2436, 3900, 5940, 8690, 12298, 16926, 22750, 29960, 38760, 49368, 62016, 76950, 94430, 114730, 138138, 164956, 195500, 230100, 269100, 312858, 361746, 416150, 476470, 543120, 616528, 697136, 785400, 881790, 986790, 1100898

Offset: 0

Johannes W. Meijer, May 06 2011

Equals the fifth right hand column of A175136.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A175136, A162148, A190049.

Related to A000332 and A091894.

[(7*n^4+58*n^3+173*n^2+218*n+96)/12: n in [0..50]]; // Vincenzo Librandi, May 07 2011

A190048 := proc(n) option remember; a(n):=(7*n^4+58*n^3+173*n^2+218*n+96)/12 end: seq(A190048(n),n=0..35);

LinearRecurrence[{5,-10,10,-5,1}, {8,46,150,370,770}, 30] (* or *) CoefficientList[Series[(8+6*x)/(1-x)^5, {x, 0, 50}], x] (* G. C. Greubel, Jan 10 2018 *)

x='x+O('x^30); Vec((8+6*x)/(1-x)^5) \\ G. C. Greubel, Jan 10 2018

for(n=0,50, print1((7*n^4 +58*n^3 +173*n^2 +218*n +96)/12, ", ")) \\ G. C. Greubel, Jan 10 2018

G.f.: (8+6*x)/(1-x)^5.
a(n) = 8*binomial(n+4,4) + 6*binomial(n+3,4).
a(n) = A091894(4,0)*binomial(n+4,4) + A091894(4,1)*binomial(n+3,4).
a(n) = (7*n^4 +58*n^3 +173*n^2 +218*n +96)/12.

A190049 Expansion of (16+24x+2x^2)/(x-1)^6.

Original entry on oeis.org

16, 120, 482, 1412, 3402, 7168, 13692, 24264, 40524, 64504, 98670, 145964, 209846, 294336, 404056, 544272, 720936, 940728, 1211098, 1540308, 1937474, 2412608, 2976660, 3641560, 4420260, 5326776, 6376230, 7584892

Offset: 0

Johannes W. Meijer, May 06 2011

Equals the sixth right hand column of A175136.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A175136, A162148, A190048.

Related to A000389 and A091894.

[(21*n^5+245*n^4+1105*n^3+2395*n^2+2474*n+960)/60: n in [0..50]]; // Vincenzo Librandi, May 07 2011

A190049 := proc(n) option remember; a(n):=(21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60 end: seq(A190049(n),n=0..27);

LinearRecurrence[{6,-15,20,-15,6,-1}, {16,120,482,1412,3402,7168}, 30] (* or *) CoefficientList[Series[(16 +24*x +2*x^2)/(1-x)^6, {x, 0, 50}], x] (* G. C. Greubel, Jan 10 2018 *)

x='x+O('x^30); Vec((16 +24*x +2*x^2)/(1-x)^6) \\ G. C. Greubel, Jan 10 2018

for(n=0,30, print1((21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60, ", ")) \\ G. C. Greubel, Jan 10 2018

G.f.: (16 +24*x +2*x^2)/(1-x)^6.
a(n) = 16*binomial(n+5,5) +24*binomial(n+4,5) +2*binomial(n+3,5).
a(n) = A091894(5,0)*binomial(n+5,5) + A091894(5,1)*binomial(n+4,5) + A091894(5,2)*binomial(n+3,5).
a(n) = (21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60.

cp's OEIS Frontend

A007564 Shifts left when INVERT transform applied thrice.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

PARI

Formula

A133156 Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials of the second kind with exponents in decreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A207538 Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A051288 Triangle read by rows: T(n,k) = number of paths of n upsteps U and n downsteps D that contain k UUDs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A068763 Irregular triangle of the Fibonacci polynomials of A011973 multiplied diagonally by the Catalan numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A162148 a(n) = n*(n+1)*(5*n+7)/6.

A162148 a(n) = n(n+1)(5*n+7)/6.

A190049 Expansion of (16+24x+2x^2)/(x-1)^6.