A106534 -id:A106534 - cp's OEIS frontend

A280470 Triangle A106534 with reversed rows.

1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 14, 19, 26, 36, 51, 42, 56, 75, 101, 137, 188, 132, 174, 230, 305, 406, 543, 731, 429, 561, 735, 965, 1270, 1676, 2219, 2950, 1430, 1859, 2420, 3155, 4120, 5390, 7066, 9285, 12235, 4862, 6292, 8151, 10571, 13726, 17846, 23236, 30302, 39587, 51822, 16796, 21658, 27950, 36101, 46672

Offset: 0

Tony Foster III, Jan 03 2017

			Fibonacci Determinant Triangle:
    1;
    1,    2;
    2,    3,    5;
    5,    7,   10,   15;
   14,   19,   26,   36,   51;
   42,   56,   75,  101,  137,  188;
  132,  174,  230,  305,  406,  543,  731;
  429,  561,  735,  965, 1270, 1676, 2219, 2950;
  ...

P. Barry, A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, page 5
A. Cvetkovi, Predrag Rajkovic, and Milos IvkoviCatalan Numbers, the Hankel Transform, and Fibonacci Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3.

Cf. A000108, A001519, A007317, A011971, A103433, A106534, A197649.

&cat [[&+[Binomial(k,j)*Catalan(n-j): j in [0..k]]: k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 07 2017

Table[Sum[Binomial[k, j] CatalanNumber[n - j], {j, 0, k}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 08 2017 *)

C(n)=binomial(2*n,n)/(n+1);
T(n,k)=sum(j=0,k,binomial(k,j)*C(n-j));
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); \\ Joerg Arndt, Jan 15 2017

T(n,k) = Sum_{j=0..k} binomial(k,j) * A000108(n-j). - Joerg Arndt, Jan 15 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

A283298 Diagonal of the Euler-Seidel matrix for the Catalan numbers.

Original entry on oeis.org

1, 3, 26, 305, 4120, 60398, 934064, 15000903, 247766620, 4182015080, 71816825856, 1250772245698, 22039796891026, 392213323252200, 7038863826811100, 127248841020380105, 2315130641074743540, 42358284517663463380, 778876539384226875800

Offset: 0

R. J. Mathar, Jul 20 2017

Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 11.

Central elements of rows in A106534, A280470.

Cf. A000108.

A000108 := n-> binomial(2*n, n)/(n+1):
A283298 := proc(n)
    add(binomial(n,i)*A000108(n+i),i=0..n) ;
end proc:
seq(A283298(n),n=0..30) ;

Table[Sum[Binomial[n, i] CatalanNumber[n + i], {i, 0, n}], {n, 0, 50}] (* Indranil Ghosh, Jul 20 2017 *)

C(n) = binomial(2*n,n)/(n+1); \\ A000108
a(n) = sum(i=0, n, binomial(n,i) * C(n+i)); \\ Michel Marcus, Nov 12 2022

from sympy import binomial, catalan
def a(n): return sum(binomial(n, i)*catalan(n + i) for i in range(n + 1))
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017

a(n) = Sum_{i=0..n} binomial(n,i) * A000108(n+i).
D-finite with recurrence 2*n*(2*n+1)*(9*n-11)*a(n) +(-711*n^3+1589*n^2-986*n+144)*a(n-1) -10*(n-1)*(9*n-2)*(2*n-3)*a(n-2)=0.
a(n) ~ 2^(2*n) * 5^(n + 3/2) / (27 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 01 2025

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A280470 Triangle A106534 with reversed rows.

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A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

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A283298 Diagonal of the Euler-Seidel matrix for the Catalan numbers.

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