A090197 -id:A090197 - 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

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

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

A384122 Array read by antidiagonals: T(n,m) is the number of minimum dominating sets in the n X m rook complement graph.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 48, 4, 1, 1, 1, 1, 5, 100, 100, 5, 1, 1, 1, 1, 6, 185, 240, 185, 6, 1, 1, 1, 1, 7, 306, 480, 480, 306, 7, 1, 1, 1, 1, 8, 469, 840, 1000, 840, 469, 8, 1, 1, 1, 1, 9, 680, 1344, 1800, 1800, 1344, 680, 9, 1, 1

Offset: 0

Andrew Howroyd, May 20 2025

For n >= 3, m >= 3, the minimum size of a dominating set is 3.

			Array begins:
===============================================
n\m | 0 1 2   3    4    5    6     7     8 ...
----+------------------------------------------
  0 | 1 1 1   1    1    1    1     1     1 ...
  1 | 1 1 1   1    1    1    1     1     1 ...
  2 | 1 1 4   3    4    5    6     7     8 ...
  3 | 1 1 3  48  100  185  306   469   680 ...
  4 | 1 1 4 100  240  480  840  1344  2016 ...
  5 | 1 1 5 185  480 1000 1800  2940  4480 ...
  6 | 1 1 6 306  840 1800 3300  5460  8400 ...
  7 | 1 1 7 469 1344 2940 5460  9114 14112 ...
  8 | 1 1 8 680 2016 4480 8400 14112 21952 ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Minimum Dominating Set.
Eric Weisstein's World of Mathematics, Rook Complement Graph.

Main diagonal is A292074.
Column 3 is A090197(n-1), n >= 4.
Column 4 is A272871(n), n >= 4.
Cf. A384121, A384123.

T(n,m) = if(n<=2||m<=2, if(n<=1||m<=1, 1, if(n==2,m)+if(m==2,n)), 4*binomial(n,2)*binomial(m,2) + 6*binomial(n,3)*binomial(m,3) + if(n==3,m) + if(m==3,n))

T(n,m) = 4*binomial(n,2)*binomial(m,2) + 6*binomial(n,3)*binomial(m,3) for n >= 4, m >= 4.
T(n,m) = T(m,n).
T(n,0) = T(n,1) = 1.

A192845 Molecular topological indices of the sun graphs.

Original entry on oeis.org

4, 56, 180, 400, 740, 1224, 1876, 2720, 3780, 5080, 6644, 8496, 10660, 13160, 16020, 19264, 22916, 27000, 31540, 36560, 42084, 48136, 54740, 61920, 69700, 78104, 87156, 96880, 107300, 118440

Offset: 1

Eric W. Weisstein, Jul 11 2011

Sun graphs are defined for n >= 3; extended to n=1 using closed form.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A090197.

List([1..40], n -> 4*n*(-3+3*n+n^2)); # G. C. Greubel, Jan 05 2019

[4*n*(-3+3*n+n^2): n in [1..40]]; // G. C. Greubel, Jan 05 2019

Table[4*n*(-3+3*n+n^2), {n,1,40}] (* G. C. Greubel, Jan 05 2019 *)
LinearRecurrence[{4,-6,4,-1},{4,56,180,400},30] (* Harvey P. Dale, Mar 02 2024 *)

vector(40, n, 4*n*(-3+3*n+n^2)) \\ G. C. Greubel, Jan 05 2019

[4*n*(-3+3*n+n^2) for n in (1..40)] # G. C. Greubel, Jan 05 2019

a(n) = 4*n*(-3 + 3*n + n^2).
a(n) = 4*A090197(n).
G.f.: 4*x*(1 + 10*x - 5*x^2)/(1-x)^4. - Colin Barker, Aug 07 2012
E.g.f.: 4*x*(1 + 6*x + x^2)*exp(x). - G. C. Greubel, Jan 05 2019

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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.

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

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A384122 Array read by antidiagonals: T(n,m) is the number of minimum dominating sets in the n X m rook complement graph.

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A192845 Molecular topological indices of the sun graphs.

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