A008550 -id:A008550 - cp's OEIS frontend

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

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

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.

Original entry on oeis.org

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

A090200 a(n) = N(7,n), where N(7,x) is the 7th Narayana polynomial.

Original entry on oeis.org

1, 429, 4279, 20071, 65445, 171481, 387739, 788019, 1476841, 2596645, 4335711, 6936799, 10706509, 16025361, 23358595, 33267691, 46422609, 63614749, 85770631, 113966295, 149442421, 193620169, 248117739, 314767651

Offset: 0

Philippe Deléham, Jan 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001263, A008550, A090198, A090199.

[n^6+21*n^5+105*n^4+175*n^3+105*n^2+21*n+1: n in [0..30]]; // G. C. Greubel, Feb 16 2021

A090200:= n-> n^6+21*n^5+105*n^4+175*n^3+105*n^2+21*n+1; seq(A090200(n), n=0..30) # G. C. Greubel, Feb 16 2021

LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,429,4279,20071,65445,171481,387739},30] (* Harvey P. Dale, Feb 10 2019 *)

a(n) = n^6+21*n^5+105*n^4+175*n^3+105*n^2+21*n+1 \\ Charles R Greathouse IV, Jan 17 2012

[n^6+21*n^5+105*n^4+175*n^3+105*n^2+21*n+1 for n in (0..30)] # G. C. Greubel, Feb 16 2021

a(n) = N(7, n) = Sum_{k>0} A001263(7, k)*n^(k-1) = n^6 + 21*n^5 + 105*n^4 + 175*n^3 + 105*n^2 + 21*n + 1.
G.f.: (1 +422*x +1297*x^2 -908*x^3 -173*x^4 +86*x^5 -5*x^6)/(1-x)^7. - Philippe Deléham, Apr 03 2013; corrected by Georg Fischer, May 02 2019
E.g.f.: (1 +428*x +1711*x^2 +1420*x^3 +380*x^4 +36*x^5 +x^6)*exp(x). - G. C. Greubel, Feb 16 2021

Corrected by T. D. Noe, Nov 09 2006

A090198 a(n) = N(5,n), where N(5,x) is the 5th Narayana polynomial.

Original entry on oeis.org

1, 42, 197, 562, 1257, 2426, 4237, 6882, 10577, 15562, 22101, 30482, 41017, 54042, 69917, 89026, 111777, 138602, 169957, 206322, 248201, 296122, 350637, 412322, 481777, 559626, 646517, 743122, 850137, 968282, 1098301, 1240962, 1397057

Offset: 0

Philippe Deléham, Jan 22 2004

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

Cf. A001263, A008550, A090199, A090200.

[n^4 +10*n^3 +20*n^2 +10*n +1: n in [0..40]]; // G. C. Greubel, Feb 16 2021

A090198:= n-> n^4 +10*n^3 +20*n^2 +10*n +1; seq(A090198(n), n=0..40) # G. C. Greubel, Feb 16 2021

LinearRecurrence[{5,-10,10,-5,1},{1,42,197,562,1257},40] (* Harvey P. Dale, Mar 06 2020 *)

a(n) = n^4+10*n^3+20*n^2+10*n+1 \\ Charles R Greathouse IV, Jan 17 2012

[n^4 +10*n^3 +20*n^2 +10*n +1 for n in (0..40)] # G. C. Greubel, Feb 16 2021

a(n) = N(5, n) = Sum_{k>0} A001263(5, k)*n^(k-1) = n^4 +10*n^3 +20*n^2 +10*n +1.
G.f.: (1 +37*x -3*x^2 -13*x^3 +2*x^4)/(1-x)^5. - Philippe Deléham, Apr 03 2013
E.g.f.: (1 +41*x +57*x^2 +16*x^3 +x^4)*exp(x). - G. C. Greubel, Feb 16 2021

Corrected by T. D. Noe, Nov 08 2006

A090199 a(n) = N(6,n), where N(6,x) is the 6th Narayana polynomial.

Original entry on oeis.org

1, 132, 903, 3304, 8925, 20076, 39907, 72528, 123129, 198100, 305151, 453432, 653653, 918204, 1261275, 1698976, 2249457, 2933028, 3772279, 4792200, 6020301, 7486732, 9224403, 11269104, 13659625, 16437876, 19649007, 23341528, 27567429

Offset: 0

Philippe Deléham, Jan 22 2004

G. C. Greubel, 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. A001263, A008550, A090198, A090200.

[(n+1)*(n^4 +14*n^3 +36*n^2 +14*n +1): n in [0..30]]; // G. C. Greubel, Feb 16 2021

Table[(n+1)*(n^4 +14*n^3 +36*n^2 +14*n +1), {n,0,30}] (* G. C. Greubel, Feb 16 2021 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,132,903,3304,8925,20076},30] (* or *) CoefficientList[Series[(1+126 x+126 x^2-154 x^3+21 x^4)/(-1+x)^6,{x,0,30}],x] (* Harvey P. Dale, Jul 24 2021 *)

a(n)=n^5+15*n^4+50*n^3+50*n^2+15*n+1 \\ Charles R Greathouse IV, Jan 17 2012

[(n+1)*(n^4 +14*n^3 +36*n^2 +14*n +1) for n in (0..30)] # G. C. Greubel, Feb 16 2021

a(n) = N(6, n)= Sum_{k>0} A001263(6, k)*n^(k-1) = n^5 + 15*n^4 + 50*n^3 + 50*n^2 + 15*n + 1.
G.f.: (1 +126*x +126*x^2 -154*x^3 +21*x^4)/(1-x)^6. - Philippe Deléham, Apr 03 2013
E.g.f.: (1 +131*x +320*x^2 +165*x^3 +25*x^4 +x^5)*exp(x). - G. C. Greubel, Feb 16 2021

Corrected generating function in Formula field. - Harvey P. Dale, Jul 24 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)

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

cp's OEIS Frontend

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

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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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A090200 a(n) = N(7,n), where N(7,x) is the 7th Narayana polynomial.

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A090198 a(n) = N(5,n), where N(5,x) is the 5th Narayana polynomial.

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A090199 a(n) = N(6,n), where N(6,x) is the 6th Narayana polynomial.

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

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