cp's OEIS Frontend

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

%I A237124 #21 May 08 2021 02:17:55
%S A237124 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,
%T A237124 297,339,249,145,69,26,7,1,1001,1133,836,497,248,103,34,8,1,3432,3861,
%U A237124 2860,1727,891,394,146,43,9,1,11934,13364,9932,6071,3211,1484,593,199,53,10,1
%N A237124 Triangle of numbers related to Catalan numbers (A000108).
%C A237124 Riordan array (1 +x +x^2*C(x)^3, x*C(x)) where C(x) is the g.f. of A000108.
%C A237124 Diagonal sums are A000108(n).
%C A237124 Row sums are T(n+1,1).
%C A237124 T(n,0) = A071724(n-1).
%C A237124 T(n,1) = A220902(n), n>=2.
%C A237124 T(n,2) = A228404(n-2), n>=4.
%C A237124 T(n+3,3) = A033434(n).
%C A237124 T(n,n) = 1.
%C A237124 T(n+1,n) = n+1.
%C A237124 T(n+2,n) = A034856(n+1).
%H A237124 G. C. Greubel, <a href="/A237124/b237124.txt">Rows n = 0..30 of the triangle, flattened</a>
%F A237124 From _Peter Bala_, Feb 18 2018: (Start)
%F A237124 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).
%F A237124 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)
%e A237124 Triangle begins:
%e A237124       1;
%e A237124       1,     1;
%e A237124       1,     2,    1;
%e A237124       3,     4,    3,    1;
%e A237124       9,    11,    8,    4,    1;
%e A237124      28,    33,   24,   13,    5,    1;
%e A237124      90,   104,   76,   43,   19,    6,   1;
%e A237124     297,   339,  249,  145,   69,   26,   7,   1;
%e A237124    1001,  1133,  836,  497,  248,  103,  34,   8,  1;
%e A237124    3432,  3861, 2860, 1727,  891,  394, 146,  43,  9,  1;
%e A237124   11934, 13364, 9932, 6071, 3211, 1484, 593, 199, 53, 10, 1;
%e A237124   ...
%t A237124 b[n_, k_]:= Binomial[2*n-k+1, n-k];
%t A237124 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]];
%t A237124 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 08 2021 *)
%o A237124 (Sage)
%o A237124 def b(n,k): return binomial(2*n-k+1, n-k)
%o A237124 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)
%o A237124 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 08 2021
%Y A237124 Cf. A000108.
%K A237124 nonn,tabl,easy
%O A237124 0,5
%A A237124 _Philippe Deléham_, Feb 03 2014