A307883 -id:A307883 - cp's OEIS frontend

A187021 Coefficient of x^n in (1 + (n+1)x + nx^2)^n.

1, 2, 13, 136, 1921, 33876, 712909, 17383584, 481003009, 14869654300, 507406003501, 18928740714192, 765897591633409, 33392080668673832, 1559976990077534253, 77717020110946293376, 4111810085670587224065, 230190619432401207833004, 13591965974806603671569101

Offset: 0

Emanuele Munarini, Mar 02 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381 (terms 0..100 from Vincenzo Librandi)
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012

Main diagonal of A307883.

Cf. A092366, A186925, A187018, A187019, A241247, A234971.

P:=PolynomialRing(Integers()); [ Coefficients((1+(n+1)*x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011

A187021:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -(n+1)/(2*sqrt(n))) );
1, seq(A187021(n), n = 1..30); # G. C. Greubel, May 31 2020
a := n -> hypergeom([-n, -n], [1], n):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020

Flatten[{1,Table[Sum[Binomial[n,k]^2*n^k,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[If[n==0, 1, Simplify[n^(n/2)*GegenbauerC[n, -n, -(n+1)/(2 Sqrt[n])]]], {n, 0, 30}] (* Emanuele Munarini, Oct 20 2016 *)

a(n):=coeff(expand((1+(n+1)*x+n*x^2)^n),x,n);
makelist(a(n),n,0,20);

{a(n)=sum(k=0,n,binomial(n,k)^2*n^k)} \\ Paul D. Hanna, Mar 29 2011

[1]+[ n^(n/2)*gegenbauer(n, -n, -(n+1)/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, May 31 2020

a(n) = [x^n] (1 + (n+1)*x + n*x^2)^n.
a(n) = n^(n/2)*GegenbauerPoly(n,-n,-(n+1)/(2*sqrt(n))). - Emanuele Munarini, Oct 20 2016
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k. - Paul D. Hanna, Mar 29 2011
a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-1) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 17 2014
a(n) = n! * [x^n] exp((n + 1)*x) * BesselI(0,2*sqrt(n)*x). - Ilya Gutkovskiy, May 31 2020
a(n) = hypergeom([-n, -n], [1], n). - Peter Luschny, Dec 22 2020

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

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2(k-1)x + ((k+1)*x)^2).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -2, 1, 1, -2, -3, 0, 1, 1, -3, -2, 11, 6, 1, 1, -4, 1, 28, 1, 0, 1, 1, -5, 6, 45, -74, -81, -20, 1, 1, -6, 13, 56, -255, -92, 141, 0, 1, 1, -7, 22, 55, -554, 477, 1324, 363, 70, 1, 1, -8, 33, 36, -959, 2376, 2689, -3656, -1791, 0, 1

Offset: 0

Seiichi Manyama, May 02 2019

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) + k*x*y). - Seiichi Manyama, Jul 11 2020

More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 + (k-r+1)*r*x*y) for any nonzero real number r. - Seiichi Manyama, Jul 22 2020

			Square array begins:
  1,   1,   1,    1,    1,    1,      1, ...
  1,   0,  -1,   -2,   -3,   -4,     -5, ...
  1,  -2,  -3,   -2,    1,    6,     13, ...
  1,   0,  11,   28,   45,   56,     55, ...
  1,   6,   1,  -74, -255, -554,   -959, ...
  1,   0, -81,  -92,  477, 2376,   6475, ...
  1, -20, 141, 1324, 2689, -804, -20195, ...

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

Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341.
Main diagonal gives A307885.
T(n,n-1) gives A335310.
Cf. A307883, A336179.

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

T(n,k) is the coefficient of x^n in the expansion of (1 - (k-1)*x - k*x^2)^n.
T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} (-k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-1) * T(n-2,k).

A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)x)/(1 - 2(k-1)x + ((k-3)x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 2, -15, 1, 4, -6, 32, 1, 6, 9, -12, 105, 1, 8, 30, 16, 30, -576, 1, 10, 57, 140, 25, 60, 105, 1, 12, 90, 384, 630, 36, -140, 5760, 1, 14, 129, 772, 2505, 2772, 49, -280, -13167, 1, 16, 174, 1328, 6430, 16008, 12012, 64, 630, -30400

Offset: 0

Seiichi Manyama, Jan 18 2020

			Square array begins:
      1,   1,  1,    1,     1,     1, ...
      0,   2,  4,    6,     8,    10, ...
    -15,  -6,  9,   30,    57,    90, ...
     32, -12, 16,  140,   384,   772, ...
    105,  30, 25,  630,  2505,  6430, ...
   -576,  60, 36, 2772, 16008, 52524, ...
.
From _Peter Luschny_, Jan 20 2020: (Start)
Read by ascending antidiagonals gives:
[0]      1
[1]      0,    1
[2]    -15,    2,  1
[3]     32,   -6,  4,     1
[4]    105,  -12,  9,     6,     1
[5]   -576,   30, 16,    30,     8,    1
[6]    105,   60, 25,   140,    57,   10,    1
[7]   5760, -140, 36,   630,   384,   90,   12,   1
[8] -13167, -280, 49,  2772,  2505,  772,  129,  14,  1
[9] -30400,  630, 64, 12012, 16008, 6430, 1328, 174, 16, 1 (End)

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

Columns k=0..5 give A331551, A331552, A000290(n+1), A002457, A108666(n+1), A331323.
T(n,n+3) gives A331512.
Cf. A307883, A331513, A331514.

T := (n, k) -> (n + 1)^2*hypergeom([-n, -n], [2], k - 2):
seq(lprint(seq(simplify(T(n,k)), k=0..7)), n=0..6) # Peter Luschny, Jan 20 2020

T[n_, k_] := (n + 1)^2 * HypergeometricPFQ[{-n, -n}, {2}, k - 2];  Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)

T(n,k) = Sum_{j=0..n} (k-3)^(n-j) * (n+j+1) * binomial(n,j) * binomial(n+j,j).
T(n,k) = Sum_{j=0..n} (k-2)^j * (j+1) * binomial(n+1,j+1)^2.
T(n,k) = (n + 1)^2*hypergeom([-n, -n], [2], k - 2). - Peter Luschny, Jan 20 2020
n * (2*n-1) * T(n,k) = 2 * (2 * (k-1) * n^2 - k + 2) * T(n-1,k) - (k-3)^2 * n * (2*n+1) * T(n-2,k) for n>1. - Seiichi Manyama, Jan 25 2020

A335333 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2(2k+1)*x + x^2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 37, 63, 1, 1, 9, 73, 305, 321, 1, 1, 11, 121, 847, 2641, 1683, 1, 1, 13, 181, 1809, 10321, 23525, 8989, 1, 1, 15, 253, 3311, 28401, 129367, 213445, 48639, 1, 1, 17, 337, 5473, 63601, 458649, 1651609, 1961825, 265729, 1

Offset: 0

Seiichi Manyama, Jun 02 2020

			Square array begins:
  1,    1,     1,      1,      1,       1, ...
  1,    3,     5,      7,      9,      11, ...
  1,   13,    37,     73,    121,     181, ...
  1,   63,   305,    847,   1809,    3311, ...
  1,  321,  2641,  10321,  28401,   63601, ...
  1, 1683, 23525, 129367, 458649, 1256651, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Legendre Polynomial.

Columns k=0..4 give A000012, A001850, A006442, A084768, A084769.
Rows n=0..6 give A000012, A005408, A003154(n+1), A160674, A144124, A335338, A144126.
Main diagonal gives A331656.
T(n,n-1) gives A331657.
Cf. A307883, A307884.

T[n_, k_] := LegendreP[n, 2*k + 1]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)

PARI
```
T(n, k) = pollegendre(n, 2*k+1);
```

T(n,k) is the coefficient of x^n in the expansion of (1 + (2*k+1)*x + k*(k+1)*x^2)^n.
T(n,k) = Sum_{j=0..n} k^j * (k+1)^(n-j) * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (n-1) * T(n-2,k).
T(n,k) = P_n(2*k+1), where P_n is n-th Legendre polynomial.
From Seiichi Manyama, Aug 30 2025: (Start)
T(n,k) = (-1)^n * Sum_{j=0..n} (1/(2*(2*k+1)))^(n-2*j) * binomial(-1/2,j) * binomial(j,n-j).
T(n,k) = Sum_{j=0..floor(n/2)} (k*(k+1))^j * (2*k+1)^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
E.g.f. of column k: exp((2*k+1)*x) * BesselI(0, 2*sqrt(k*(k+1))*x). (End)

A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)x) / (1-2(k+1)x+((k-1)x)^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 4, 17, 32, 1, 1, 5, 28, 99, 128, 1, 1, 6, 41, 208, 577, 512, 1, 1, 7, 56, 365, 1552, 3363, 2048, 1, 1, 8, 73, 576, 3281, 11584, 19601, 8192, 1, 1, 9, 92, 847, 6016, 29525, 86464, 114243, 32768, 1, 1, 10, 113, 1184, 10033, 62976, 265721, 645376, 665857, 131072, 1

Offset: 0

Seiichi Manyama, Sep 04 2020

			Square array begins:
  1,   1,    1,     1,     1,     1, ...
  1,   2,    3,     4,     5,     6, ...
  1,   8,   17,    28,    41,    56, ...
  1,  32,   99,   208,   365,   576, ...
  1, 128,  577,  1552,  3281,  6016, ...
  1, 512, 3363, 11584, 29525, 62976, ...

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

Column k=0..9 give A000012, A081294, A001541, A090965, A083884, A099140, A099141, A099142, A165224, A026244.
Main diagonal gives A333990.
Cf. A009999, A307883, A337389, A333989.

T[n_, 0] := 1; T[n_, k_] := Sum[k^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 04 2020 *)

{T(n, k) = sum(j=0, n, k^j*binomial(2*n, 2*j))}

T(n,k) = Sum_{j=0..n} k^j * binomial(2*n,2*j).

T(0,k) = 1, T(1,k) = k+1 and T(n,k) = 2 * (k+1) * T(n-1,k) - (k-1)^2 * T(n-2,k) for n>1.

A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2kx + ((k-2)*x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 3, -6, 1, 6, 6, 0, 1, 9, 30, 10, 30, 1, 12, 66, 140, 15, 0, 1, 15, 114, 450, 630, 21, -140, 1, 18, 174, 1000, 2955, 2772, 28, 0, 1, 21, 246, 1850, 8430, 18963, 12012, 36, 630, 1, 24, 330, 3060, 18855, 69384, 119812, 51480, 45, 0

Offset: 0

Seiichi Manyama, Jan 19 2020

			Square array begins:
    1,  1,    1,     1,     1,      1, ...
    0,  3,    6,     9,    12,     15, ...
   -6,  6,   30,    66,   114,    174, ...
    0, 10,  140,   450,  1000,   1850, ...
   30, 15,  630,  2955,  8430,  18855, ...
    0, 21, 2772, 18963, 69384, 187425, ...

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

Columns k=1..5 give A000217(n+1), A002457, A002695(n+1), A331515, A331516.

Cf. A307883, A331511.

T[n_, k_] = 1/2 * Sum[If[k == 2 && n == j - 1, 1, (k - 2)^(n + 1 - j)] * j * Binomial[n + 1, j] * Binomial[n + 1 + j, j], {j, 1, n + 1}]; Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)

T(n,k) = (1/2)*sum(j=1,n+1,(k-2)^(n+1-j)*j*binomial(n+1,j)*binomial(n+1+j,j));
matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jan 20 2020

T(n,k) = (1/2) * Sum_{j=1..n+1} (k-2)^(n+1-j) * j * binomial(n+1,j) * binomial(n+1+j,j).
n * T(n,k) = k * (2*n+1) * T(n-1,k) - (k-2)^2 * (n+1) * T(n-2,k) for n > 1.
T(n,k) = ((n+2)/2) * Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
T(n,k) = Sum_{j=0..n} (k/2)^j * (-(k-2)^2/(2*k))^(n-j) * (2*j+1) * binomial(2*j,j) * binomial(j,n-j) for k > 0. - Seiichi Manyama, Aug 20 2025

A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2kx + ((k-2)x)^2 + (1 - kx) * sqrt(1 - 2kx + ((k-2)*x)^2)).

Original entry on oeis.org

1, 1, 0, 1, 2, -3, 1, 4, 3, 0, 1, 6, 15, 4, 10, 1, 8, 33, 56, 5, 0, 1, 10, 57, 180, 210, 6, -35, 1, 12, 87, 400, 985, 792, 7, 0, 1, 14, 123, 740, 2810, 5418, 3003, 8, 126, 1, 16, 165, 1224, 6285, 19824, 29953, 11440, 9, 0, 1, 18, 213, 1876, 12130, 53550, 140497, 166344, 43758, 10, -462

Offset: 0

Seiichi Manyama, Jan 26 2020

			Square array begins:
   1, 1,   1,    1,     1,     1, ...
   0, 2,   4,    6,     8,    10, ...
  -3, 3,  15,   33,    57,    87, ...
   0, 4,  56,  180,   400,   740, ...
  10, 5, 210,  985,  2810,  6285, ...
   0, 6, 792, 5418, 19824, 53550, ...

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

Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.
T(n,n+1) gives A331794.
Cf. A307883, A331511, A331514, A331795.

T[n_, k_] := Sum[If[k==1 && j==0, 1, (k-1)^j] * Binomial[n + 1, j] * Binomial[n + 1, j + 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 2021 *)

T(n,k) = Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
n * (n+2) * T(n,k) = (n+1) * (k * (2*n+1) * T(n-1,k) - (k-2)^2 * n * T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (k-1)^j * k^(n-2*j) * binomial(n+1,n-2*j) * binomial(2*j+1,j). - Seiichi Manyama, Aug 24 2025
From Seiichi Manyama, Aug 27 2025: (Start)
T(n,k) = [x^n] (1+k*x+(k-1)*x^2)^(n+1).
For k != 1, e.g.f. of column k: exp(k*x) * BesselI(1, 2*sqrt(k-1)*x) / sqrt(k-1), with offset 1. (End)

A341014 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * j! * binomial(n,j)^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 17, 34, 1, 1, 5, 31, 139, 209, 1, 1, 6, 49, 352, 1473, 1546, 1, 1, 7, 71, 709, 5233, 19091, 13327, 1, 1, 8, 97, 1246, 13505, 95836, 291793, 130922, 1, 1, 9, 127, 1999, 28881, 318181, 2080999, 5129307, 1441729, 1

Offset: 0

Seiichi Manyama, Feb 02 2021

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     3,     4,      5,      6, ...
  1,    7,    17,    31,     49,     71, ...
  1,   34,   139,   352,    709,   1246, ...
  1,  209,  1473,  5233,  13505,  28881, ...
  1, 1546, 19091, 95836, 318181, 830126, ...

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

Columns 0..4 give A000012, A002720, A025167, A102757, A102773.
Rows 0..2 give A000012, A000027(n+1), A056220(n+1).
Main diagonal gives A330260.
Cf. A307883.

T[n_, k_] := Sum[If[j == k == 0, 1, k^j]*j!*Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 02 2021 *)

{T(n,k) = sum(j=0, n, k^j*j!*binomial(n, j)^2)}

E.g.f. of column k: exp(x/(1-k*x)) / (1-k*x).

T(n,k) = (2*k*n-k+1) * T(n-1,k) - k^2 * (n-1)^2 * T(n-2,k) for n > 1.

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 7, 0, 1, 4, 15, 32, 19, 0, 1, 5, 24, 81, 136, 51, 0, 1, 6, 35, 160, 459, 592, 141, 0, 1, 7, 48, 275, 1120, 2673, 2624, 393, 0, 1, 8, 63, 432, 2275, 8064, 15849, 11776, 1107, 0, 1, 9, 80, 637, 4104, 19375, 59136, 95175, 53344, 3139, 0

Offset: 0

Seiichi Manyama, May 05 2019

			Square array begins:
   1,   1,     1,     1,      1,       1,       1, ...
   0,   1,     2,     3,      4,       5,       6, ...
   0,   3,     8,    15,     24,      35,      48, ...
   0,   7,    32,    81,    160,     275,     432, ...
   0,  19,   136,   459,   1120,    2275,    4104, ...
   0,  51,   592,  2673,   8064,   19375,   40176, ...
   0, 141,  2624, 15849,  59136,  168125,  400896, ...
   0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

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

Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.
Main diagonal gives A092366.
Cf. A107267, A292627, A307819, A307847, A307855, A307883.

A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - k * (k-4) * (n-1) * A(n-2,k).

cp's OEIS Frontend

A187021 Coefficient of x^n in (1 + (n+1)*x + n*x^2)^n.

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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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A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2).

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A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)*x)/(1 - 2*(k-1)*x + ((k-3)*x)^2)^(3/2).

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A335333 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*(2*k+1)*x + x^2).

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A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)*x) / (1-2*(k+1)*x+((k-1)*x)^2).

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A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2*k*x + ((k-2)*x)^2)^(3/2).

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A187021 Coefficient of x^n in (1 + (n+1)x + nx^2)^n.

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2(k-1)x + ((k+1)*x)^2).

A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)x)/(1 - 2(k-1)x + ((k-3)x)^2)^(3/2).

A335333 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2(2k+1)*x + x^2).

A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)x) / (1-2(k+1)x+((k-1)x)^2).

A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2kx + ((k-2)*x)^2)^(3/2).

A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2kx + ((k-2)x)^2 + (1 - kx) * sqrt(1 - 2kx + ((k-2)*x)^2)).

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).