A059231 -id:A059231 - cp's OEIS frontend

A243631 Square array of Narayana polynomials N_n evaluated at the integers, A(n,k) = N_n(k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 11, 14, 1, 1, 1, 5, 19, 45, 42, 1, 1, 1, 6, 29, 100, 197, 132, 1, 1, 1, 7, 41, 185, 562, 903, 429, 1, 1, 1, 8, 55, 306, 1257, 3304, 4279, 1430, 1, 1, 1, 9, 71, 469, 2426, 8925, 20071, 20793, 4862, 1

Offset: 0

Peter Luschny, Jun 08 2014

Mirror image of A008550. - Philippe Deléham, Sep 26 2014

			   [0]  [1]      [2]      [3]      [4]      [5]      [6]     [7]
[0] 1,   1,       1,       1,       1,       1,       1,       1
[1] 1,   1,       1,       1,       1,       1,       1,       1
[2] 1,   2,       3,       4,       5,       6,       7,       8 .. A000027
[3] 1,   5,      11,      19,      29,      41,      55,      71 .. A028387
[4] 1,  14,      45,     100,     185,     306,     469,     680 .. A090197
[5] 1,  42,     197,     562,    1257,    2426,    4237,    6882 .. A090198
[6] 1, 132,     903,    3304,    8925,   20076,   39907,   72528 .. A090199
[7] 1, 429,    4279,   20071,   65445,  171481,  387739,  788019 .. A090200
   A000108, A001003, A007564, A059231, A078009, A078018, A081178
First few rows of the antidiagonal triangle are:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2,  1;
  1, 1, 3,  5,  1;
  1, 1, 4, 11, 14,  1;
  1, 1, 5, 19, 45, 42, 1; - _G. C. Greubel_, Feb 16 2021

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Cf. A001263, A008550 (mirror), A204057 (another version), A242369 (main diagonal), A099169 (diagonal), A307883, A336727.
Rows[2-7]: A000027, A028387, A090197, A090198, A090199, A090200.
Columns[1-7]: A000108, A001003, A007564, A059231, A078009, A078018, A081178.
Cf. A132745.

A243631:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n,j)^2*k^j*(n-j)/(n*(j+1)): j in [0..n-1]]) >;
[A243631(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

# Computed with Narayana polynomials:
N := (n,k) -> binomial(n,k)^2*(n-k)/(n*(k+1));
A := (n,x) -> `if`(n=0, 1, add(N(n,k)*x^k, k=0..n-1));
seq(print(seq(A(n,k), k=0..7)), n=0..7);
# Computed by recurrence:
Prec := proc(n,N,k) option remember; local A,B,C,h;
if n = 0 then 1 elif n = 1 then 1+N+(1-N)*(1-2*k)
else h := 2*N-n; A := n*h*(1+N-n); C := n*(h+2)*(N-n);
B := (1+h-n)*(n*(1-2*k)*(1+h)+2*k*N*(1+N));
(B*Prec(n-1,N,k) - C*Prec(n-2,N,k))/A fi end:
T := (n, k) -> Prec(n,n,k)/(n+1);
seq(print(seq(T(n,k), k=0..7)), n=0..7);
# Array by o.g.f. of columns:
gf := n -> 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1):
for n from 0 to 11 do PolynomialTools:-CoefficientList(convert( series(gf(n), x, 12), polynom), x) od; # Peter Luschny, Nov 17 2014
# Row n by linear recurrence:
rec := n -> a(x) = add((-1)^(k+1)*binomial(n,k)*a(x-k), k=1..n):
ini := n -> seq(a(k) = A(n,k), k=0..n): # for A see above
row := n -> gfun:-rectoproc({rec(n),ini(n)},a(x),list):
for n from 1 to 7 do row(n)(8) od; # Peter Luschny, Nov 19 2014

MatrixForm[Table[JacobiP[n,1,-2*n-1,1-2*x]/(n+1), {n,0,7},{x,0,7}]]
Table[Hypergeometric2F1[1-k, -k, 2, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

def NarayanaPolynomial():
    R = PolynomialRing(ZZ, 'x')
    D = [1]
    h = 0
    b = True
    while True:
        if b :
            for k in range(h, 0, -1):
                D[k] += x*D[k-1]
            h += 1
            yield R(expand(D[0]))
            D.append(0)
        else :
            for k in range(0, h, 1):
                D[k] += D[k+1]
        b = not b
NP = NarayanaPolynomial()
for _ in range(8):
    p = next(NP)
    [p(k) for k in range(8)]

def A243631(n,k): return 1 if n==0 else sum( binomial(n,j)^2*k^j*(n-j)/(n*(j+1)) for j in [0..n-1])
flatten([[A243631(k,n-k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Feb 16 2021

T(n, k) = 2F1([1-n, -n], [2], k), 2F1 the hypergeometric function.
T(n, k) = P(n,1,-2*n-1,1-2*k)/(n+1), P the Jacobi polynomials.
T(n, k) = sum(j=0..n-1, binomial(n,j)^2*(n-j)/(n*(j+1))*k^j), for n>0.
For a recurrence see the second Maple program.
The o.g.f. of column n is gf(n) = 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1). - Peter Luschny, Nov 17 2014
T(n, k) ~ (sqrt(k)+1)^(2*n+1)/(2*sqrt(Pi)*k^(3/4)*n^(3/2)). - Peter Luschny, Nov 17 2014
The n-th row can for n>=1 be computed by a linear recurrence, a(x) = sum(k=1..n, (-1)^(k+1)*binomial(n,k)*a(x-k)) with initial values a(k) = p(n,k) for k=0..n and p(n,x) = sum(j=0..n-1, binomial(n-1,j)*binomial(n,j)*x^j/(j+1)) (implemented in the fourth Maple script). - Peter Luschny, Nov 19 2014
(n+1) * T(n,k) = (k+1) * (2*n-1) * T(n-1,k) - (k-1)^2 * (n-2) * T(n-2,k) for n>1. - Seiichi Manyama, Aug 08 2020
Sum_{k=0..n} T(k, n-k) = Sum_{k=0..n} 2F1([-k, 1-k], [2], n-k) = A132745(n). - G. C. Greubel, Feb 16 2021

A059450 Triangle read by rows: T(n,k) = Sum_{j=0..k-1} T(n,j) + Sum_{j=1..n-k} T(n-j,k), with T(0,0)=1 and T(n,k) = 0 for k > n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 4, 8, 17, 29, 8, 20, 50, 107, 185, 16, 48, 136, 336, 721, 1257, 32, 112, 352, 968, 2370, 5091, 8925, 64, 256, 880, 2640, 7116, 17304, 37185, 65445, 128, 576, 2144, 6928, 20168, 53596, 129650, 278635, 491825, 256, 1280, 5120, 17664, 54880

Offset: 0

N. J. A. Sloane, Sep 16 2003

G.f. A(x,y) satisfies 0 = -(1-x)^2 + (1-x)(1-4x+3xy)A + 2x(1-2x-2y+3xy)A^2. G.f.: (1-x)(-(1-4x+3xy) + sqrt((1-xy)(1-9xy)))/(4x(1-2x-2y+3xy)) = 2(1-x)/(1-4x+3xy+sqrt((1-xy)(1-9xy))). - Michael Somos, Mar 06 2004
T(n,k) = number of below-diagonal lattice paths from (0,0) to (n,k) consisting of steps (k,0) (k=1,2,...) and (0,k) (k=1,2,...). Example: T(2,1)=3 because we have (1,0)(1,0)(0,1), (2,0)(0,1) and (1,0)(0,1)(1,0). - Emeric Deutsch, Mar 19 2004
T(n,k) is odd if and only if (n,k) = (0,0), k = n > 0, or k + 1 = n > 0. - Peter Kagey, Apr 20 2020

			1;
1,  1;
2,  3,  5;
4,  8, 17,  29;
8, 20, 50, 107, 185;

Wen-jin Woan, Diagonal lattice paths, Congressus Numerantium, 151, 2001, 173-178.

Peter Kagey, Table of n, a(n) for n = 0..10011 (141 rows flattened, first 50 rows from G. C. Greubel)
C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.

Columns include A000079, A001792 (I guess), A086866, A059231. Rows sums give A086871.

A059231(n) = T(n, n).

l := 1:a[0,0] := 1:b[l] := 1:T := (n,k)->sum(a[n,j],j=0..k-1)+sum(a[n-j,k],j=1..n-k): for n from 1 to 15 do for k from 0 to n do a[n,k] := T(n,k):l := l+1:b[l] := a[n,k]: od:od:seq(b[w],w=1..l); # Sascha Kurz
# alternative
A059450 := proc(n,k)
    option remember;
    local j ;
    if k =0 and n= 0 then
        1;
    elif k > n or k < 0 then
        0 ;
    else
        add( procname(n,j),j=0..k-1) + add(procname(n-j,k),j=1..n-k) ;
    end if;
end proc:
seq(seq(A059450(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Mar 25 2024

t[0, 0] = 1; t[n_, k_] /; k > n = 0; t[n_, k_] := t[n, k] = Sum[t[n, j], {j, 0, k-1}] + Sum[t[n-j, k], {j, 1, n-k}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

T(n,k)=if(k<0||k>n,0,polcoeff(polcoeff(2*(1-x)/((1-4*x+3*x*y)+sqrt((1-x*y)*(1-9*x*y)+x^2*O(x^n))),n),k)) /* Michael Somos, Mar 06 2004 */

T(n,k)=local(A,t);if(k<0||k>n,0,A=matrix(n+1,n+1);A[1,1]=1;for(m=1,n,t=0;for(j=0,m,t+=(A[m+1,j+1]=t+sum(i=1,m-j,A[m-i+1,j+1]))));A[n+1,k+1]) /* Michael Somos, Mar 06 2004 */

T(n,k)=if(k<0||k>n,0,(n==0)+sum(j=0,k-1,T(n,j))+sum(j=1,n-k,T(n-j,k))) /* Michael Somos, Mar 06 2004 */

More terms from Ray Chandler, Sep 17 2003

A082147 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^kN(n,k) where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

Original entry on oeis.org

1, 1, 9, 89, 945, 10577, 123129, 1476841, 18130401, 226739489, 2878666857, 37006326777, 480750990993, 6301611631473, 83240669582937, 1106980509493641, 14808497812637121, 199138509770855489, 2690461489090104009

Offset: 0

Benoit Cloitre, May 10 2003

nonn

More generally coefficients of (1 + m*x - sqrt(m^2*x^2 - (2*m+4)*x+1))/( (2*m+2)*x) are given by a(n) = Sum_{k=0..n} (m+1)^k*N(n,k)).
The Hankel transform of this sequence is 8^C(n+1,2). - Philippe Deléham, Oct 29 2007
Shifts left when INVERT transform applied eight times. - Benedict W. J. Irwin, Feb 07 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A001003, A007564, A059231.

a:=n->Sum([0..n],k->8^k*(1/n)*Binomial(n,k)*Binomial(n,k+1));;
Concatenation([1],List([1..18],n->a(n))); # Muniru A Asiru, Feb 10 2018

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1+7*x-Sqrt(49*x^2-18*x+1))/(16*x))) // G. C. Greubel, Feb 05 2018

A082147_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]+8*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A082147_list(18); # Peter Luschny, May 19 2011

Table[SeriesCoefficient[(1+7*x-Sqrt[49*x^2-18*x+1])/(16*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
f[n_] := Sum[ 8^k*Binomial[n, k]*Binomial[n, k + 1]/n, {k, 0, n}]; f[0] = 1; Array[f, 21, 0] (* Robert G. Wilson v, Feb 24 2018 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 8];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

a(n)=if(n<1,1,sum(k=0,n,8^k/n*binomial(n,k)*binomial(n,k+1)))

G.f.: (1 + 7*x - sqrt(49*x^2-18*x+1))/(16*x).
a(n) = Sum_{k=0..n} A088617(n, k)*8^k*(-7)^(n-k). - Philippe Deléham, Jan 21 2004
a(n) = (9(2n-1)a(n-1) - 49(n-2)a(n-2)) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  8, 8, 8
  1, 1, 1, 1
  8, 8, 8, 8, 8
  1, 1, 1, 1, 1, 1
  ...
- Gary W. Adamson, Jul 08 2011
a(n) ~ sqrt(16+18*sqrt(2))*(9+4*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - x/(1 - 8*x/(1 - x/(1 - 8*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017
a(n) = hypergeom([1 - n, -n], [2], 8). - Peter Luschny, Mar 19 2018

A131198 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,1,0,1,0,1,0,...] DELTA [0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0

Offset: 0

Philippe Deléham, Oct 20 2007

Mirror image of triangle A090181, another version of triangle of Narayana (A001263).

Equals A133336*A130595 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007

			Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  3,   1,   0;
  1,  6,   6,   1,   0;
  1, 10,  20,  10,   1,   0;
  1, 15,  50,  50,  15,   1,  0;
  1, 21, 105, 175, 105,  21,  1, 0;
  1, 28, 196, 490, 490, 196, 28, 1, 0; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), Article 09.7.6.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path., The number of double rises of a Dyck path., The number of valleys of a Dyck path., The number of left oriented leafs except the first one of a binary tree., The number of left tunnels of a Dyck path.
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.

Cf. A000217, A002415, A006542, A006857.

[[n le 0 select 1 else (n-k)*Binomial(n,k)^2/(n*(k+1)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

T := (n,k) -> `if`(n=0, 0^n, binomial(n,k)^2*(n-k)/(n*(k+1)));
seq(print(seq(T(n,k), k=0..n)), n=0..5); # Peter Luschny, Jun 08 2014
R := n -> simplify(hypergeom([1 - n, -n], [2], x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022

Table[If[n == 0, 1, (n-k)*Binomial[n,k]^2/(n*(k+1))], {n,0,10}, {k,0,n}] //Flatten (* G. C. Greubel, Feb 06 2018 *)

for(n=0,10, for(k=0,n, print1(if(n==0,1, (n-k)*binomial(n,k)^2/(n* (k+1))), ", "))) \\ G. C. Greubel, Feb 06 2018

Sum_{k=0..n} T(n,k)*x^k = A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Oct 23 2007
Sum_{k=0..floor(n/2)} T(n-k,k) = A004148(n). - Philippe Deléham, Nov 06 2007
T(2*n,n) = A125558(n). - Philippe Deléham, Nov 16 2011
T(n, k) = [x^k] hypergeom([1 - n, -n], [2], x). - Peter Luschny, Apr 26 2022

A082148 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 10^kN(n,k), where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

Original entry on oeis.org

1, 1, 11, 131, 1661, 22101, 305151, 4335711, 63009881, 932449961, 14004694451, 212944033051, 3271618296661, 50711564152381, 792088104593511, 12454801769554551, 196991734871121201, 3131967533789345361, 50026642742943415131, 802406215117502069811

Offset: 0

Benoit Cloitre, May 10 2003

nonn

More generally, coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/((2*m+2)*x) are given by: a(n) = Sum_{k=0..n} (m+1)^k*N(n,k)).
The Hankel transform of this sequence is 10^C(n+1,2). - Philippe Deléham, Oct 29 2007
a(n) = upper left term in M^n, M = the production matrix:
   1,  1;
  10, 10, 10;
   1,  1,  1,  1;
  10, 10, 10, 10, 10;
   1,  1,  1,  1,  1,  1;
  ...
- Gary W. Adamson, Jul 08 2011
Shifts left when INVERT transform applied ten times. - Benedict W. J. Irwin, Feb 07 2016
For fixed m > 0, if g.f. = (1+m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/((2*m+2)*x) then a(n,m) ~ (m + 2 + 2*sqrt(m+1))^(n + 1/2) / (2*sqrt(Pi) * (m+1)^(3/4) * n^(3/2)). - Vaclav Kotesovec, Mar 19 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A001003, A007564, A059231.

I:=[1,11]; [1] cat [n le 2 select I[n] else (11*(2*n-1)*Self(n-1) - 81*(n-2)*Self(n-2))/(n+1): n in [1..30]]; // G. C. Greubel, Feb 10 2018

A082148_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]+10*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A082148_list(17); # Peter Luschny, May 19 2011

Table[SeriesCoefficient[(1+9*x-Sqrt[81*x^2-22*x+1])/(20*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Sum[10^k*1/n*Binomial[n, k]*Binomial[n, k + 1], {k, 0, n}];
a[0] = 1; Array[a, 20, 0] (* Robert G. Wilson v, Feb 10 2018 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 10];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

a(n)=if(n<1,1,sum(k=0,n,10^k/n*binomial(n,k)*binomial(n,k+1)))

G.f.: (1+9*x-sqrt(81*x^2-22*x+1))/(20*x).
a(n) = Sum_{k=0..n} A088617(n, k)*10^k*(-9)^(n-k). - Philippe Deléham, Jan 21 2004
a(n) = (11*(2n-1)*a(n-1) - 81*(n-2)*a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - Philippe Deléham, Aug 19 2005
a(n) ~ sqrt(20+11*sqrt(10))*(11+2*sqrt(10))^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - x/(1 - 10*x/(1 - x/(1 - 10*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017
a(n) = hypergeom([1 - n, -n], [2], 10). - Peter Luschny, Mar 19 2018

A082173 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 11^kN(n,k) where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

Original entry on oeis.org

1, 1, 12, 155, 2124, 30482, 453432, 6936799, 108507180, 1727970542, 27924685416, 456820603086, 7550600079672, 125905525750500, 2115511349837040, 35782547891727495, 608787760350045420, 10411451736723707990

Offset: 0

Benoit Cloitre, May 10 2003

nonn

More generally coefficients of (1 + m*x - sqrt(m^2*x^2 -(2*m+4)*x + 1) )/((2*m+2)*x) are given by a(n) = Sum_{k=0..n} (m+1)^k*N(n,k).
The Hankel transform of this sequence is 11^C(n+1,2). - Philippe Deléham, Oct 29 2007
For fixed m > 0, if g.f. = (1 + m*x - sqrt(m^2*x^2 -(2*m+4)*x + 1) )/((2*m+2)*x) then a(n,m) ~ (m + 2 + 2*sqrt(m+1))^(n + 1/2) / (2*sqrt(Pi) * (m+1)^(3/4) * n^(3/2)). - Vaclav Kotesovec, Mar 19 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001003, A001263, A007564, A059231, A088617.

[1] cat [&+[11^k*Binomial(n, k)*Binomial(n, k+1)/n:k in [0..n]]:n in [1..18]]; // Marius A. Burtea, Jan 22 2020

A082173_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]+11*add(a[j]*a[w-j-1],j=1..w-1)od;
convert(a, list) end: A082173_list(17); # Peter Luschny, May 19 2011

Table[SeriesCoefficient[(1+10*x-Sqrt[100*x^2-24*x+1])/(22*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 11];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

a(n)=if(n<1,1,sum(k=0,n,11^k/n*binomial(n,k)*binomial(n,k+1)))

def A082173_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+10*x-sqrt(100*x^2-24*x+1))/(22*x) ).list()
A082173_list(30) # G. C. Greubel, Jan 21 2024

G.f.: (1+10*x-sqrt(100*x^2-24*x+1))/(22*x).
a(n) = Sum_{k=0..n} A088617(n, k)*11^k*(-10)^(n-k). - Philippe Deléham, Jan 21 2004
a(n) = (12*(2n-1)*a(n-1) - 100*(n-2)*a(n-2)) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
   1,  1
  11, 11, 11
   1,  1,  1,  1
  11, 11, 11, 11, 11
   1,  1,  1,  1,  1,  1
  ... (End)
a(n) ~ sqrt(22+12*sqrt(11))*(12+2*sqrt(11))^n/(22*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - x/(1 - 11*x/(1 - x/(1 - 11*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017
a(n) = hypergeom([1 - n, -n], [2], 11). - Peter Luschny, Mar 19 2018

A082181 a(0) = 1, for n>=1, a(n) = Sum_{k=0..n} 9^kN(n,k), where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

Original entry on oeis.org

1, 1, 10, 109, 1270, 15562, 198100, 2596645, 34825150, 475697854, 6595646860, 92590323058, 1313427716380, 18798095833012, 271118225915560, 3936516861402901, 57494017447915150, 844109420603623030

Offset: 0

Benoit Cloitre, May 10 2003

nonn

More generally, coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/((2*m+2)*x) are given by: a(n) = Sum_{k=0..n} (m+1)^k*N(n,k).
The Hankel transform of this sequence is 9^C(n+1,2). - Philippe Deléham, Oct 29 2007
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  9, 9, 9
  1, 1, 1, 1
  9, 9, 9, 9, 9
  1, 1, 1, 1, 1, 1
  ... (End)
Shifts left when INVERT transform applied nine times. - Benedict W. J. Irwin, Feb 07 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A001003, A001263, A007564, A059231, A088617.

[(&+[Binomial(n,k)*Binomial(n-1,k)*9^k/(k+1): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 23 2022

A082181_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]+9*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A082181_list(17); # Peter Luschny, May 19 2011

Table[SeriesCoefficient[(1+8*x-Sqrt[64*x^2-20*x+1])/(18*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 9];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

a(n)=if(n<1,1,sum(k=0,n,9^k/n*binomial(n,k)*binomial(n,k+1)))

[sum(binomial(n,k)*binomial(n-1,k)*9^k/(k+1) for k in (0..n)) for n in (0..30)] # G. C. Greubel, May 23 2022

G.f.: (1+8*x-sqrt(64*x^2-20*x+1))/(18*x).
a(n) = Sum_{k=0..n} A088617(n, k)*9^k*(-8)^(n-k). - Philippe Deléham, Jan 21 2004
a(n) = (10*(2*n-1)*a(n-1) - 64*(n-2)*a(n-2)) / (n+1) for n>=2, a(0)=a(1)=1. - Philippe Deléham, Aug 19 2005
a(n) ~ 2^(4*n+1)/(3*sqrt(3*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - x/(1 - 9*x/(1 - x/(1 - 9*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017
a(n) = hypergeom([1 - n, -n], [2], 9). - Peter Luschny, Mar 19 2018

Corrected by T. D. Noe, Oct 25 2006

A127846 Series reversion of x/(1+5x+4x^2).

Original entry on oeis.org

0, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785

Offset: 0

Paul Barry, Feb 02 2007

Hankel transform is -A127847(n)=-4^C(n,2)*(4^n-1)/3; a(n+1) counts (5,4)-Motzkin paths of length n, where there are 5 colors available for the H(1,0) steps and 4 for the U(1,1) steps. See A059231 for more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A059231

CoefficientList[Series[(1-5*x-Sqrt[1-10*x+9*x^2])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

A127846 = lambda n: hypergeometric([1-n, -n], [2], 4) if n>0 else 0
[Integer(A127846(n).n(100)) for n in (0..22)] # Peter Luschny, Sep 23 2014

G.f.: (1-5x-sqrt(1-10x+9x^2))/(8x); a(n)=sum{k=0..n-1, (1/n)*C(n,k)C(n,k+1)4^k}; a(n+1)=sum{k=0..floor(n/2), C(n, 2k)C(k)5^(n-2k)*4^k};
Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 3^(2*n+1)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
a(n) = A059231(n) for n>0. - Philippe Deléham, Apr 03 2013
a(n) = hypergeom([1-n, -n], [2], 4) for n>0. - Peter Luschny, Sep 23 2014

A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 5, 1, 0, 14, 21, 9, 1, 0, 42, 84, 56, 14, 1, 0, 132, 330, 300, 120, 20, 1, 0, 429, 1287, 1485, 825, 225, 27, 1, 0, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 0, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 0, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1, 0

Offset: 0

Philippe Deléham, Oct 19 2007

Mirror image of triangle A086810; another version of A126216.
Equals A131198*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007
Diagonal sums: A119370. - Philippe Deléham, Nov 09 2009

			Triangle begins:
    1;
    1,    0;
    2,    1,    0;
    5,    5,    1,   0;
   14,   21,    9,   1,   0;
   42,   84,   56,  14,   1,  0;
  132,  330,  300, 120,  20,  1, 0;
  429, 1287, 1485, 825, 225, 27, 1, 0;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

[[Binomial(n-1,k)*Binomial(2*n-k,n)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[Binomial[n-1,k]*Binomial[2*n-k,n]/(n+1), {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 05 2018 *)

for(n=0,10, for(k=0,n, print1(binomial(n-1,k)*binomial(2*n-k,n)/(n+1), ", "))) \\ G. C. Greubel, Feb 05 2018

Sum_{k=0..n} T(n,k)*x^k = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - Philippe Deléham, Nov 02 2009

A331516 Expansion of 1/(1 - 10x + 9x^2)^(3/2).

Original entry on oeis.org

1, 15, 174, 1850, 18855, 187425, 1832460, 17705700, 169569405, 1612842275, 15256106778, 143660483070, 1347716324227, 12603114069525, 117536416879320, 1093553079352200, 10153324144411065, 94098595671581175, 870667876141568070, 8044341506669534850

Offset: 0

Seiichi Manyama, Jan 19 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column 5 of A331514.

Cf. A059231, A084771, A383946.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1 - 10*x + 9*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020

[1/2*&+[3^(n-k+1)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..20]]; // Marius A. Burtea, Jan 20 2020

a[n_] := 1/2 * Sum[3^( n + 1 - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n+1}]; Array[a, 20, 0] (* Amiram Eldar, Jan 20 2020 *)
CoefficientList[Series[1/(1-10x+9x^2)^(3/2),{x,0,20}],x] (* Harvey P. Dale, Nov 04 2021 *)

my(N=20, x='x+O('x^N)); Vec(1/(1-10*x+9*x^2)^(3/2))

a(n) = sum(k=1, n+1, 3^(n+1-k)*k*binomial(n+1, k)*binomial(n+1+k, k))/2;

a(n) = 1/2 * Sum_{k=1..n+1} 3^(n+1-k) * k * binomial(n+1,k) * binomial(n+1+k,k).
n * a(n) = 5 * (2*n+1) * a(n-1) - 9 * (n+1) * a(n-2) for n > 1.
a(n) = ((n+2)/2) * Sum_{k=0..n} 4^k * binomial(n+1,k) * binomial(n+1,k+1).
a(n) ~ sqrt(n) * 3^(2*n + 3) / (2^(7/2) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A059231(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (5/2)^k * (-9/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)

cp's OEIS Frontend

A243631 Square array of Narayana polynomials N_n evaluated at the integers, A(n,k) = N_n(k), n>=0, k>=0, read by antidiagonals.

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A059450 Triangle read by rows: T(n,k) = Sum_{j=0..k-1} T(n,j) + Sum_{j=1..n-k} T(n-j,k), with T(0,0)=1 and T(n,k) = 0 for k > n.

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A082147 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

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A131198 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,1,0,1,0,1,0,...] DELTA [0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.

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A082148 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 10^k*N(n,k), where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

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A082173 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 11^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

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A082147 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 8^kN(n,k) where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

A082148 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 10^kN(n,k), where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

A082173 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 11^kN(n,k) where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

A082181 a(0) = 1, for n>=1, a(n) = Sum_{k=0..n} 9^kN(n,k), where N(n,k) = (1/n)C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

A331516 Expansion of 1/(1 - 10x + 9x^2)^(3/2).