A098332 -id:A098332 - cp's OEIS frontend

A335310 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * (-n)^(n-k).

1, 1, -2, 11, -74, 477, -804, -84425, 3315334, -102211207, 3005297956, -88338323709, 2627003399164, -78764141488043, 2341929797646648, -66394419743289105, 1609460569459689286, -18001777147777896975, -1625299659961386724524, 196005371138608184827003

Offset: 0

Ilya Gutkovskiy, May 31 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A098332, A116091, A126869, A307884, A307885, A331657, A335309.

Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] (-n)^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[SeriesCoefficient[1/Sqrt[1 + 2 (n - 2) x + n^2 x^2], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[(2 - n) x] BesselI[0, 2 Sqrt[1 - n] x], {x, 0, n}], {n, 0, 19}]
Table[Hypergeometric2F1[-n, -n, 1, 1 - n], {n, 0, 19}]

a(n) = sum(k=0, n, binomial(n,k)^2*(1-n)^k); \\ Michel Marcus, Jun 01 2020

a(n) = central coefficient of (1 - (n - 2)*x - (n - 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 + 2*(n - 2)*x + n^2*x^2).
a(n) = n! * [x^n] exp((2 - n)*x) * BesselI(0,2*sqrt(1 - n)*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (1-n)^k.

A098339 Expansion of 1/sqrt(1 - 6x + 17x^2).

Original entry on oeis.org

1, 3, 5, -9, -111, -477, -1051, 1095, 21793, 106947, 276165, -71145, -4712655, -26071965, -76452315, -29748249, 1045547073, 6564746115, 21507513221, 19922192439, -230801512751, -1674387214173, -6072718662555

Offset: 0

Paul Barry, Sep 03 2004

Binomial transform of A098336. Second binomial transform of A098332.

Central coefficients of (1 + 3x - 2x^2)^n.

Michael De Vlieger, Table of n, a(n) for n = 0..1628
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

CoefficientList[Series[1/Sqrt[1-6x+17x^2],{x,0,30}],x] (* Harvey P. Dale, Jun 19 2013 *)

E.g.f.: exp(3x)*BesselI(0, 2*sqrt(-2)*x).

D-finite with recurrence: n*a(n) + 3*(1-2*n)*a(n-1) + 17*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 09 2012

A336201 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 0, 1, 1, -2, -3, 0, 1, 1, -3, -14, 11, 0, 1, 1, -4, -47, 136, 1, 0, 1, 1, -5, -134, 909, 106, -81, 0, 1, 1, -6, -347, 4736, 3585, -8492, 141, 0, 1, 1, -7, -846, 21655, 61906, -323523, 35344, 363, 0, 1, 1, -8, -1983, 91512, 771601, -8065624, 2201809, 395008, -1791, 0, 1

Offset: 0

Seiichi Manyama, Jul 11 2020

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) + k * Product_{j=1..k} x_j) for k>0.

			Square array begins:
  1, 1,   1,     1,       1,        1, ...
  1, 0,  -1,    -2,      -3,       -4, ...
  1, 0,  -3,   -14,     -47,     -134, ...
  1, 0,  11,   136,     909,     4736, ...
  1, 0,   1,   106,    3585,    61906, ...
  1, 0, -81, -8492, -323523, -8065624, ...

Columns k=0-3 give: A000012, A000007, (-1)^n*A098332(n), A336182.
Main diagonal gives A336202.
Cf. A309010, A336179, A336187.

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

cp's OEIS Frontend

A335310 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * (-n)^(n-k).

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A098339 Expansion of 1/sqrt(1 - 6x + 17x^2).

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A336201 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^k.

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