A026012 -id:A026012 - cp's OEIS frontend

A228339 Fourth differences of Catalan numbers (A000108).

3, 9, 30, 102, 352, 1230, 4344, 15483, 55626, 201246, 732564, 2681223, 9861342, 36428320, 135100620, 502841295, 1877678370, 7032454470, 26410804020, 99437742720, 375260126904, 1419223506516, 5378236459328, 20419260060462, 77659985060772, 295844249258796, 1128738495397128, 4312685074680465, 16500218817839274, 63209983347693924

Offset: 0

N. J. A. Sloane, Aug 29 2013

nonn

Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000108, A000245, A026012, A033434, A059346.

Differences[Table[CatalanNumber[n], {n, 0, 30}], 4] (* Amiram Eldar, Jul 10 2023 *)

From Amiram Eldar, Jul 10 2023: (Start)
a(n) = 9*(9*n^4 + 54*n^3 + 135*n^2 + 122*n + 40) * n! * binomial(2*n, n)/(n+5)!.
Sum_{n>=0} a(n)/4^n = 38. (End)

A382668 a(n) = C(n+1) - C(n-1) - 2*C(n-2) where C(n) = A000108(n) are the Catalan numbers.

Original entry on oeis.org

2, 10, 33, 108, 359, 1214, 4169, 14508, 51064, 181492, 650522, 2348856, 8535921, 31197430, 114601065, 422891340, 1566903060, 5827192140, 21743726430, 81383916840, 305465105790, 1149489049644, 4335921660522, 16391329697528, 62091796219904, 235656705875304

Offset: 2

F. Chapoton, Apr 02 2025

Cf. A000108, A026012, A280891.

gf := ((2*x^3 + x^2 - 1)*sqrt(1 - 4*x) - 4*x^3 - 3*x^2 - 2*x + 1)/(2*x^2):
ser := series(gf, x, 30): seq(coeff(ser, x, n), n = 2..27);  # Peter Luschny, Apr 03 2025

a[n_]:=CatalanNumber[n+1]-CatalanNumber[n-1]-2CatalanNumber[n-2];Array[a,26,2] (* James C. McMahon, Apr 05 2025 *)

C = catalan_number
[C(n + 1) - C(n - 1) - 2 * C(n - 2) for n in range(2, 28)]

a(n) = [x^n] ((2*x^3 + x^2 - 1)*sqrt(1 - 4*x) - 4*x^3 - 3*x^2 - 2*x + 1)/(2*x^2). - Peter Luschny, Apr 03 2025

cp's OEIS Frontend

A228339 Fourth differences of Catalan numbers (A000108).

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