A033434 -id:A033434 - cp's OEIS frontend

A283439 Hankel transform of A033434.

1, -3, -9, -6, 10, 25, 15, -21, -49, -28, 36, 81, 45, -55, -121, -66, 78, 169, 91, -105, -225, -120, 136, 289, 153, -171, -361, -190, 210, 441, 231, -253, -529, -276, 300, 625, 325, -351, -729, -378, 406, 841, 435, -465, -961, -496, 528, 1089, 561, -595, -1225

Offset: 0

Paul Barry, Mar 07 2017

a(n) modulo 2 is A131719(n+2).

Cf. A131719, A033434.

CoefficientList[Series[(1 - 5*x + 2*x^3 - 3*x^4 + 2*x^5 - x^6)/((1 + x)*(1 - x + x^2)^3) ,{x, 0, 35}], x] (* Indranil Ghosh, Mar 08 2017 *)

print(Vec((1 - 5*x + 2*x^3 - 3*x^4 + 2*x^5 - x^6)/((1 + x)*(1 - x + x^2)^3) + O(x^36))); \\ Indranil Ghosh, Mar 08 2017

G.f.: (1 - 5*x + 2*x^3 - 3*x^4 + 2*x^5 - x^6)/((1 + x)*(1 - x + x^2)^3).
a(3*k)     =  (-1)^k*(k + 1)*(2*k + 1).
a(3*k + 1) = -(-1)^k*(k + 1)*(2*k + 3).
a(3*k + 2) = -(-1)^k*(k + 3)^2.

More terms from Indranil Ghosh, Mar 08 2017

A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 3, 5, 3, 4, 6, 9, 14, 6, 9, 13, 19, 28, 42, 15, 21, 30, 43, 62, 90, 132, 36, 51, 72, 102, 145, 207, 297, 429, 91, 127, 178, 250, 352, 497, 704, 1001, 1430, 232, 323, 450, 628, 878, 1230, 1727, 2431, 3432, 4862, 603, 835, 1158, 1608, 2236, 3114

Offset: 0

N. J. A. Sloane, Jan 27 2001

			Array starts:
      1       1       2       5      14      42     132     429
      0       1       3       9      28      90     297    1001
      1       2       6      19      62     207     704    2431
      1       4      13      43     145     497    1727    6071
      3       9      30     102     352    1230    4344   15483
      6      21      72     250     878    3114   11139   40143
     15      51     178     628    2236    8025   29004  105477
     36     127     450    1608    5789   20979   76473  280221
     91     323    1158    4181   15190   55494  203748  751422
    232     835    3023   11009   40304  148254  547674 2031054
    603    2188    7986   29295  107950  399420 1483380 5527750
Triangle starts:
  1;
  0,  1;
  1,  1,  2;
  1,  2,  3,  5;
  3,  4,  6,  9, 14;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
Zhousheng Mei, Suijie Wang, Pattern Avoidance of Generalized Permutations, arXiv:1804.06265 [math.CO], 2018.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See p. 36.

Top row is A000108, leading diagonals give A005043, A001006, A005554.
Row sums are A106640.
Cf. A000108, A000245, A026012, A033434, A106534.

T := (n,k) -> (-1)^(n-k)*binomial(2*k,k)*hypergeom([k-n,k+1/2], [k+2], 4)/(k+1): seq(seq(simplify(T(n,k)), k=0..n), n=0..10);
# Peter Luschny, Aug 16 2012, updated May 25 2021

max = 11; t = Table[ Differences[ Table[ CatalanNumber[k], {k, 0, max}], n], {n, 0, max}]; Flatten[ Table[t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011 *)

def T(n, k) :
    if k > n : return 0
    if n == k : return binomial(2*n, n)/(n+1)
    return T(n-1, k) - T(n, k+1)
A059346 = lambda n,k: (-1)^(n-k)*T(n, k)
for n in (0..5): [A059346(n,k) for k in (0..n)] # Peter Luschny, Aug 16 2012

T(n, k) = (-1)^(n-k)*binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2],[k+2], 4). - Peter Luschny, Aug 16 2012

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001

A237124 Triangle of numbers related to Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 3, 1, 9, 11, 8, 4, 1, 28, 33, 24, 13, 5, 1, 90, 104, 76, 43, 19, 6, 1, 297, 339, 249, 145, 69, 26, 7, 1, 1001, 1133, 836, 497, 248, 103, 34, 8, 1, 3432, 3861, 2860, 1727, 891, 394, 146, 43, 9, 1, 11934, 13364, 9932, 6071, 3211, 1484, 593, 199, 53, 10, 1

Offset: 0

Philippe Deléham, Feb 03 2014

Riordan array (1 +x +x^2*C(x)^3, x*C(x)) where C(x) is the g.f. of A000108.
Diagonal sums are A000108(n).
Row sums are T(n+1,1).
T(n,0) = A071724(n-1).
T(n,1) = A220902(n), n>=2.
T(n,2) = A228404(n-2), n>=4.
T(n+3,3) = A033434(n).
T(n,n) = 1.
T(n+1,n) = n+1.
T(n+2,n) = A034856(n+1).

			Triangle begins:
      1;
      1,     1;
      1,     2,    1;
      3,     4,    3,    1;
      9,    11,    8,    4,    1;
     28,    33,   24,   13,    5,    1;
     90,   104,   76,   43,   19,    6,   1;
    297,   339,  249,  145,   69,   26,   7,   1;
   1001,  1133,  836,  497,  248,  103,  34,   8,  1;
   3432,  3861, 2860, 1727,  891,  394, 146,  43,  9,  1;
  11934, 13364, 9932, 6071, 3211, 1484, 593, 199, 53, 10, 1;
  ...

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A000108.

b[n_, k_]:= Binomial[2*n-k+1, n-k];
T[n_, k_]:= If[n<3, Binomial[n, k], b[n, k] -2*b[n, k+1] -b[n, k+2] +3*b[n, k+3] - 2*b[n, k+4]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 08 2021 *)

def b(n,k): return binomial(2*n-k+1, n-k)
def T(n,k): return binomial(n,k) if (n<3) else b(n,k) -2*b(n, k+1) -b(n, k+2) +3*b(n, k+3) -2*b(n, k+4)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021

From Peter Bala, Feb 18 2018: (Start)
T(n,k) = C(2*n+1-k, n-k) - 2*C(2*n-k, n-k-1) - C(2*n-1-k, n-k-2) + 3*C(2*n-2-k, n-k-3) - 2*C(2*n-3-k, n-k-4), for n > 2, otherwise C(n, k).
The n-th row polynomial of the row reverse triangle equals the n-th degree Taylor polynomial of the function (1 - x^2 + x^3)*(1 - 2*x)/(1 - x)^2 * 1/(1 - x)^n about 0. For example, for n = 4, (1 - x^2 + x^3)*(1 - 2*x)/(1 - x)^2 * 1/(1 - x)^4 = 1 + 4*x + 8*x^2 + 11*x^3 + 9*x^4 + O(x^5), giving (9, 11, 8, 4, 1) as row 4. (End)

A228339 Fourth differences of Catalan numbers (A000108).

Original entry on oeis.org

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)

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A283439 Hankel transform of A033434.

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A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

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A237124 Triangle of numbers related to Catalan numbers (A000108).

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A228339 Fourth differences of Catalan numbers (A000108).

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