A204057 -id:A204057

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A008550

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

+10
9

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

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

LINKS

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).

FORMULA

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

EXAMPLE

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, 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

MAPLE

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

MATHEMATICA

(* 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 *)

PROG

(Sage) 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

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

CROSSREFS

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.

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jan 23 2004

STATUS

approved

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.

+10
9

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

[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, 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

MAPLE

# 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

MATHEMATICA

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 *)

PROG

(Sage)

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)]

(Sage)

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

(Magma)

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

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jun 08 2014

STATUS

approved

A099169

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A diagonal of Narayana array (A008550).

LINKS

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

FORMULA

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)

MAPLE

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

MATHEMATICA

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 *)

PROG

(Sage)

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

(Magma)

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

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

CROSSREFS

Cf. A092366, A187018, A187019, A187021.

Cf. A008550, A204057, A243631.

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan, Oct 09 2004

STATUS

approved

A242369

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

MAPLE

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);

MATHEMATICA

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

PROG

(Sage)

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

(Magma)

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

CROSSREFS

Main diagonal of A008550, A243631.

Cf. A204057.

KEYWORD

nonn

AUTHOR

Peter Luschny, Jun 08 2014

STATUS

approved

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