A243631 -id:A243631 - cp's OEIS frontend

A008550 Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial.

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

Offset: 0

Philippe Deléham, Jan 23 2004

Mirror image of A243631. - Philippe Deléham, Sep 26 2014

			Row n=0:  1, 1,  1,   1,    1,     1,      1, ... see A000012.
Row n=1:  1, 1,  2,   5,   14,    42,    132, ... see A000108.
Row n=2:  1, 1,  3,  11,   45,   197,    903, ... see A001003.
Row n=3:  1, 1,  4,  19,  100,   562,   3304, ... see A007564.
Row n=4:  1, 1,  5,  29,  185,  1257,   8925, ... see A059231.
Row n=5:  1, 1,  6,  41,  306,  2426,  20076, ... see A078009.
Row n=6:  1, 1,  7,  55,  469,  4237,  39907, ... see A078018.
Row n=7:  1, 1,  8,  71,  680,  6882,  72528, ... see A081178.
Row n=8:  1, 1,  9,  89,  945, 10577, 123129, ... see A082147.
Row n=9:  1, 1, 10, 109, 1270, 15562, 198100, ... see A082181.
Row n=10: 1, 1, 11, 131,  161,  1661,  22101, ... see A082148.
Row n=11: 1, 1, 12, 155, 2124, 30482, 453432, ... see A082173.
... - _Philippe Deléham_, Apr 03 2013
The first few rows of the antidiagonal triangle are:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  5,  3,  1, 1;
  1, 14, 11,  4, 1, 1;
  1, 42, 45, 19, 5, 1, 1; - _G. C. Greubel_, Feb 15 2021

Michael De Vlieger, Table of n, a(n) for n = 0..11475
H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, arXiv:math/0103149 [math.CO], 2001.
H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, Séminaire Lotharingien de Combinatoire 46 (2001), Article B46a.
L. Yang, S.-L. Yang, A relation between Schroder paths and Motzkin paths, Graphs Combinat. 36 (2020) 1489-1502, eq. (6).

Columns: A000012, A000012, A000027, A028387, A090197, A090198, A090199, A090200.
Main diagonal is A242369.
A diagonal is in A099169.
Cf. A204057 (another version), A088617, A243631.
Cf. A132745.

[Truncate(HypergeometricSeries(k-n, k-n+1, 2, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 15 2021

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

(* First program *)
Unprotect[Power]; Power[0 | 0, 0 | 0] = 1; Protect[Power]; Table[Function[n, Sum[Apply[Binomial[#1 + #2, #1] Binomial[#1, #2]/(#2 + 1) &, {k, j}]*n^j*(1 - n)^(k - j), {j, 0, k}]][m - k + 1] /. k_ /; k <= 0 -> 1, {m, -1, 9}, {k, m + 1, 0, -1}] // Flatten (* Michael De Vlieger, Aug 10 2017 Note: this code renders 0^0 = 1. To restore normal Power functionality: Unprotect[Power]; ClearAll[Power]; Protect[Power] *)
(* Second program *)
Table[Hypergeometric2F1[1-n+k, k-n, 2, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)

flatten([[hypergeometric([k-n, k-n+1], [2], k).simplify_hypergeometric() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 15 2021

T(n, k) = Sum_{j>0} A001263(k, j)*n^(j-1); T(n, 0)=1.
T(n, k) = Sum_{j, 0<=j<=k} A088617(k, j)*n^j*(1-n)^(k-j).
The o.g.f. of row 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
G.f. of row n: 1/(1 - x/(1 - n*x/(1 - x/(1 - n*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017
T(n, k) = Hypergeometric2F1([k-n, k-n+1], [2], k), as a number triangle. - G. C. Greubel, Feb 15 2021

A204057 Triangle derived from an array of f(x), Narayana polynomials.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 14, 1, 1, 5, 19, 45, 42, 1, 1, 6, 29, 100, 197, 132, 1, 1, 7, 41, 185, 562, 903, 429, 1, 1, 8, 55, 306, 1257, 3304, 4279, 1430, 1, 1, 9, 71, 469, 2426, 8925, 20071, 20793, 4862, 1, 1, 10, 89, 680, 4237, 20076, 65445, 124996, 103049, 16796, 1

Offset: 1

Gary W. Adamson, Jan 09 2012

Row sums = (1, 2, 4, 10, 31, 113, 466, 2129, 10641, 138628, 335379, 2702364,...)
Another version of triangle in A008550. - Philippe Deléham, Jan 13 2012
Another version of A243631. - Philippe Deléham, Sep 26 2014

			First few rows of the array =
  1,....1,....1,.....1,.....1,...; = A000012
  1.....2,....5,....14,....42,...; = A000108
  1,....3,...11,....45,...197,...; = A001003
  1,....4,...19,...100,...562,...; = A007564
  1,....5,...29,...185,..1257,...; = A059231
  1,....6,...41,...306,..2426,...; = A078009
  ...
First few rows of the triangle =
  1;
  1, 1;
  1, 2,  1;
  1, 3,  5,   1;
  1, 4, 11,  14,    1;
  1, 5, 19,  45,   42,    1;
  1, 6, 29, 100,  197,  132,     1;
  1, 7, 41, 185,  562,  903,   429,     1;
  1, 8, 55, 306, 1257, 3304,  4279,  1430,    1;
  1, 9, 71, 469, 2426, 8952, 20071, 20793, 4862, 1;
  ...
Examples: column 4 of the array = A090197: (1, 14, 45, 100,...) = N(4,n) where N(4,x) is the 4th Narayana polynomial.
Term (5,3) = 29 is the upper left term of M^3, where M = the infinite square production matrix:
  1, 4, 0, 0, 0,...
  1, 1, 4, 0, 0,...
  1, 1, 1, 4, 0,...
  1, 1, 1, 1, 4,...
... generating row 5, A059231: (1, 5, 29, 185,...).

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000108, A001003, A007564, A028387, A059231, A078009, A090197, A090198, A090199, A090200.

Cf. A008550, A132745, A243631.

A204057:= 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]]) >;
[A204057(k, n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 16 2021

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

def A204057(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([[A204057(k, n-k) for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Feb 16 2021

The triangle is the set of antidiagonals of an array in which columns are f(x) of the Narayana polynomials; with column 1 = (1, 1, 1,...) column 2 = (1, 2, 3,..), column 3 = A028387, column 4 = A090197, then A090198, A090199,...
The array by rows is generated from production matrices of the form:
  1, (N-1)
  1, 1, (N-1)
  1, 1, 1, (N-1)
  1, 1, 1, 1, (N-1)
...(infinite square matrices with the rest zeros); such that if the matrix is M, n-th term in row N is the upper left term of M^n.
From G. C. Greubel, Feb 16 2021: (Start)
T(n, k) = Hypergeometric2F1([1-k, -k], [2], n-k).
Sum_{k=1..n} T(n, k) = A132745(n) - 1. (End)

Corrected by Philippe Deléham, Jan 13 2012

A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.

Original entry on oeis.org

1, 2, 11, 100, 1257, 20076, 387739, 8766248, 226739489, 6595646860, 212944033051, 7550600079672, 291527929539433, 12169325847587832, 545918747361417291, 26183626498897556176, 1336713063706757646465

Offset: 1

Ralf Stephan, Oct 09 2004

A diagonal of Narayana array (A008550).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012

Cf. A092366, A187018, A187019, A187021.

Cf. A008550, A204057, A243631.

A099169:= func< n | (&+[Binomial(n, j)*Binomial(n-1,j)*(n-1)^j/(j+1): j in [0..n-1]]) >;
[A099169(n): n in [1..30]]; // G. C. Greubel, Feb 16 2021

A099169:= n-> add( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1), j=0..n-1);
seq( A099169(n), n=1..30) # G. C. Greubel, Feb 16 2021

Join[{1},Table[Sum[Binomial[n,k]Binomial[n,k+1](n-1)^k,{k,0,n-1}]/n,{n,2,20}]] (* Harvey P. Dale, Oct 07 2013 *)
Table[Hypergeometric2F1[1-n,-n,2,-1+n],{n,1,20}] (* Vaclav Kotesovec, Apr 18 2014 *)

a(n) = (1/n) * sum(k=0, n-1, binomial(n,k) * binomial(n,k+1) * (n-1)^k); \\ Michel Marcus, Feb 16 2021

def A099169(n): return sum( binomial(n, j)*binomial(n-1,j)*(n-1)^j/(j+1) for j in [0..n-1])
[A099169(n) for n in [1..30]] # G. C. Greubel, Feb 16 2021

From Vaclav Kotesovec, Apr 18 2014, extended Dec 01 2021: (Start)
a(n) = Hypergeometric2F1([1-n,-n], [2], -1+n).
a(n) ~ exp(2*sqrt(n)-2) * n^(n-7/4) / (2*sqrt(Pi)) * (1 + 119/(48*sqrt(n))). (End)

A132745 Row sums of (A008550 formatted as a triangular array).

Original entry on oeis.org

1, 2, 3, 5, 11, 32, 114, 467, 2130, 10642, 57629, 335381, 2082582, 13716502, 95352529, 696790819, 5334094259, 42649956716, 355261078352, 3075741331481, 27620835538407, 256810928552476, 2468108094076860, 24481671811988907, 250296546308500181, 2634309876797453868, 28509045368598994348

Offset: 0

Philippe Deléham, Nov 21 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A001263, A008550, A243631.

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]]) >;
A132745:= func< n | (&+[A243631(k,n-k): k in [0..n]]) >;
[A132745(n): n in [0..30]]; // G. C. Greubel, Feb 16 2021

Table[Sum[Hypergeometric2F1[1-k, -k, 2, n-k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 16 2021 *)

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])
def A132745(n): return sum( A243631(k, n-k) for k in [0..n] )
[A132745(n) for n in [0..30]] # G. C. Greubel, Feb 16 2021

From G. C. Greubel, Feb 16 2021: (Start)
a(n) = Sum_{k=0..n} Hypergeometric2F1([1-n+k, k-n], [2], k).
a(n) = Sum_{k=0..n} Hypergeometric2F1([1-k, -k], [2], n-k).
a(n) = 1 + Sum_{k=1..n} Sum_{j=0..k-1} binomial(k,j)^2 * ((k-j)*(n-k)^j/(k*(j+1))).
a(n) = 1 + Sum_{k=1..n} Sum_{j=0..k-1} A001263(k, k-j) * (n-k)^j. (End)

Terms a(11) onward added by G. C. Greubel, Feb 16 2021

A242369 a(n) = P(n, 1, -2n-1, 1-2n)/(n+1), P the Jacobi polynomial.

Original entry on oeis.org

1, 1, 3, 19, 185, 2426, 39907, 788019, 18130401, 475697854, 14004694451, 456820603086, 16343563014649, 636020474595988, 26736885607750515, 1207031709414024451, 58225055056545820545, 2988064457570991780854, 162517551565531508113699, 9336340704734213892357498

Offset: 0

Peter Luschny, Jun 08 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350

Main diagonal of A008550, A243631.

Cf. A204057.

A242369:= func< n | n eq 0 select 1 else (&+[Binomial(n, j)^2*(n-j)*n^(j-1)/(j+1): j in [0..n-1]]) >;
[A242369(n): n in [0..30]]; // G. C. Greubel, Feb 16 2021

a := n -> `if`(n=0,1, add(binomial(n,j)^2*(n-j)/(j+1)*n^(j-1), j=0..n-1)); seq(a(n), n=0..20);

Table[JacobiP[n, 1, -2*n-1, 1-2*n]/(n+1), {n, 0, 20}]

def A242369(n): return 1 if n==0 else sum( binomial(n, j)^2*(n-j)*n^(j-1)/(j+1) for j in [0..n-1])
[A242369(n) for n in [0..20]] # G. C. Greubel, Feb 16 2021

a(n) = 2F1([1-n, -n], [2], n), 2F1 the hypergeometric function.
a(n) = Sum_{j=0..n-1} ( binomial(n,j)^2*(n-j)/(j+1)*n^(j-1) ), for n>0.
a(n) ~ (sqrt(n)+1)^(2*n+1)/(2*sqrt(Pi)*(n+1/2)^(9/4)). - Peter Luschny, Nov 17 2014

A336727 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-k)^(n-j) * binomial(n,j) * binomial(n,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 0, 1, 1, 1, -3, 1, 5, 2, 1, 1, 1, -4, 5, 10, -3, 0, 1, 1, 1, -5, 11, 9, -38, -21, -5, 1, 1, 1, -6, 19, -4, -103, 28, 51, 0, 1, 1, 1, -7, 29, -35, -174, 357, 289, 41, 14, 1, 1, 1, -8, 41, -90, -203, 1176, -131, -1262, -391, 0, 1

Offset: 0

Seiichi Manyama, Aug 02 2020

			  1,  1,   1,   1,    1,    1,    1, ...
  1,  1,   1,   1,    1,    1,    1, ...
  1,  0,  -1,  -2,   -3,   -4,   -5, ...
  1, -1,  -1,   1,    5,   11,   19, ...
  1,  0,   5,  10,    9,   -4,  -35, ...
  1,  2,  -3, -38, -103, -174, -203, ...
  1,  0, -21,  28,  357, 1176, 2575, ...

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

Columns k=0-3 give: A000012, A090192, (-1)^n * A154825(n), A336729.
Main diagonal gives A336728.
Cf. A243631, A307884, A336708, A336709.

T[0, k_] := 1; T[n_, k_] := Sum[If[k == 0, Boole[n == j],(-k)^(n - j)] * Binomial[n, j] * Binomial[n , j - 1], {j, 1, n}] / n; Table[T[k, n- k], {n, 0, 11}, {k, 0, n}] //Flatten (* Amiram Eldar, Aug 02 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, (-k)^(n-j)*binomial(n, j)*binomial(n, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+k*x*A)); polcoef(A, n)}

{T(n, k) = sum(j=0, n, (-k)^j*(k+1)^(n-j)*binomial(n, j)*binomial(n+j, n)/(j+1))}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x) / (1 + k * x * A_k(x)).
A_k(x) = 2/(1 - (k+1)*x + sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2)).
T(n, k) = Sum_{j=0..n} (-k)^j * (k+1)^(n-j) * binomial(n,j) * binomial(n+j,n)/(j+1).
(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

A247507 Square array read by ascending antidiagonals, n>=0, k>=0. Row n is the expansion of (1-nx-sqrt(n^2x^2-2nx-4x+1))/(2x).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 22, 14, 1, 5, 20, 57, 90, 42, 1, 6, 30, 116, 300, 394, 132, 1, 7, 42, 205, 740, 1686, 1806, 429, 1, 8, 56, 330, 1530, 5028, 9912, 8558, 1430, 1, 9, 72, 497, 2814, 12130, 35700, 60213, 41586, 4862

Offset: 0

Peter Luschny, Nov 17 2014

			   [0][1] [2]  [3]    [4]     [5]      [6]       [7]
[0] 1, 1,  2,   5,    14,     42,     132,      429,.. A000108
[1] 1, 2,  6,  22,    90,    394,    1806,     8558,.. A006318
[2] 1, 3, 12,  57,   300,   1686,    9912,    60213,.. A047891
[3] 1, 4, 20, 116,   740,   5028,   35700,   261780,.. A082298
[4] 1, 5, 30, 205,  1530,  12130,  100380,   857405,.. A082301
[5] 1, 6, 42, 330,  2814,  25422,  239442,  2326434,.. A082302
[6] 1, 7, 56, 497,  4760,  48174,  507696,  5516133,.. A082305
[7] 1, 8, 72, 712,  7560,  84616,  985032, 11814728,.. A082366
[8] 1, 9, 90, 981, 11430, 140058, 1782900, 23369805,.. A082367

L. Yang, S.-L. Yang, A relation between Schroder paths and Motzkin paths, Graphs Combinat. 36 (2020) 1489-1502, eq. (5).

Cf. A243631.

Main diagonal gives A302286.

gf := n -> (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x):
for n from 0 to 10 do lprint(PolynomialTools:-CoefficientList( convert(series(gf(n),x,8),polynom),x)) od;

G.f. of row n: 1/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 06 2018

Offset changed to 0 by Alois P. Heinz, May 28 2015

A247502 Triangle read by rows: coefficients of polynomials related to the exponential generating function of sequences generated by Narayana polynomials evaluated at the integers; n>=1, 0<=k

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 9, 1, 1, 41, 57, 16, 1, 1, 131, 320, 165, 25, 1, 1, 428, 1711, 1420, 380, 36, 1, 1, 1429, 8967, 11151, 4620, 756, 49, 1, 1, 4861, 46663, 83202, 49665, 12306, 1358, 64, 1, 1, 16795, 242634, 602407, 495327, 172893, 28476, 2262, 81, 1

Offset: 1

Peter Luschny, Nov 18 2014

Definition: Let N(n,x) = Sum_{j=0..n-1} x^j*C(n,j)^2*(n-j)/(n*(j+1)) for n>0 and N(0,x) = 1, further let p(n,x) be implicitly defined by N(n,k) = k!*[x^k](exp(x)*p(n,x)), then T(n,k) = [x^k] p(n,x).

			Triangle T(n,k) begins:
[n\k][0,    1,     2,     3,     4,     5,    6,  8, 9]
[1]   1,
[2]   1,    1,
[3]   1,    4,     1,
[4]   1,   13,     9,     1,
[5]   1,   41,    57,    16,     1,
[6]   1,  131,   320,   165,    25,     1,
[7]   1,  428,  1711,  1420,   380,    36,    1,
[8]   1, 1429,  8967, 11151,  4620,   756,   49,  1,
[9]   1, 4861, 46663, 83202, 49665, 12306, 1358, 64, 1.
.
The sequence N(7,k) = 1 + 21*k + 105*k^2 + 175*k^3 + 105*k^4 + 21*k^5 + k^6 = 1, 429, 4279, 20071, 65445, ... = A090200(k) has the exponential generating function exp(x)*(1 + 428*x + 1711*x^2 + 1420*x^3 + 380*x^4 + 36*x^5 + x^6). Thus T(7,3) = 1420.

Cf. A243631 and the crossreferences given there.

T(n, 0) = T(n, n-1) = 1.
T(n, 1) = A001453(n) = A000108(n) - 1 for n>=2.
T(n, n-2) = (n-1)^2 for n>=2.

cp's OEIS Frontend

A008550 Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial.

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A204057 Triangle derived from an array of f(x), Narayana polynomials.

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A099169 a(n) = (1/n) * Sum_{k=0..n-1} C(n,k) * C(n,k+1) * (n-1)^k.

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A132745 Row sums of (A008550 formatted as a triangular array).

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A242369 a(n) = P(n, 1, -2*n-1, 1-2*n)/(n+1), P the Jacobi polynomial.

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A336727 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-k)^(n-j) * binomial(n,j) * binomial(n,j-1) for n > 0.

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Mathematica

PARI

PARI

PARI

Formula

A242369 a(n) = P(n, 1, -2n-1, 1-2n)/(n+1), P the Jacobi polynomial.

A247507 Square array read by ascending antidiagonals, n>=0, k>=0. Row n is the expansion of (1-nx-sqrt(n^2x^2-2nx-4x+1))/(2x).