A106534 - OEIS

A106534

Sum array of Catalan numbers (A000108) read by upward antidiagonals.

%I #45 Jan 12 2024 10:04:58

%S 1,2,1,5,3,2,15,10,7,5,51,36,26,19,14,188,137,101,75,56,42,731,543,

%T 406,305,230,174,132,2950,2219,1676,1270,965,735,561,429,12235,9285,

%U 7066,5390,4120,3155,2420,1859,1430,51822,39587,30302,23236,17846,13726,10571,8151,6292,4862

%N Sum array of Catalan numbers (A000108) read by upward antidiagonals.

%C The underlying array A is A(n, k) = Sum_{j=0..n} binomial(n, j)*A000108(k+j), n >= 0, k>= 0. See the example section. - _Wolfdieter Lang_, Oct 04 2019

%H G. C. Greubel, <a href="/A106534/b106534.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry2/barry94r.html">The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences</a>, J. Int. Seq. 13 (2010) # 10.8.2, page 5.

%F T(n, k) = 0 if k > n; T(n, n) = A000108(n); T(n, k) = T(n-1, k) + T(n, k+1) if 0 <= k < n.

%F T(n, k) = binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2], [k+2], -4). - _Peter Luschny_, Aug 16 2012

%F T(n, k) = A(n-k, k) = Sum_{j=0..n-k} binomial(n-k, j)*A000108(k+j), n >= 0, k = 0..n. - _Wolfdieter Lang_, Oct 03 2019

%F G.f.: (sqrt(1-4*x*y)-sqrt((5*x-1)/(x-1)))/(2*x*(x*y-y+1)). - _Vladimir Kruchinin_, Jan 12 2024

%e From _Wolfdieter Lang_, Oct 04 2019: (Start)

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0: 1

%e 1: 2 1

%e 2: 5 3 2

%e 3: 15 10 7 5

%e 4: 51 36 26 19 14

%e 5: 188 137 101 75 56 42

%e 6: 731 543 406 305 230 174 132

%e 7: 2950 2219 1676 1270 965 735 561 429

%e 8: 12235 9285 7066 5390 4120 3155 2420 1859 1430

%e 9: 51822 39587 30302 23236 17846 13726 10571 8151 6292 4862

%e 10: 223191 171369 131782 101480 78244 60398 46672 36101 27950 21658 16796

%e ... reformatted and extended.

%e -------------------------------------------------------------------------

%e The array A(n, k) begins:

%e n\k 0 1 2 3 4 5 6 ...

%e -------------------------------------------

%e 0: 1 1 2 5 14 42 132 ... A000108

%e 1 2 3 7 19 56 174 561 ... A005807

%e 2: 5 10 26 75 230 735 2420 ...

%e 3: 15 36 101 305 965 3155 10571 ...

%e 4: 51 137 406 1270 4120 13726 46672 ...

%e 5: 188 543 1676 5390 17846 60398 207963 ...

%e ... (End)

%p # Uses floating point, precision might have to be adjusted.

%p C := n -> binomial(2*n,n)/(n+1);

%p H := (n,k) -> hypergeom([k-n,k+1/2],[k+2],-4);

%p T := (n,k) -> C(k)*H(n,k);

%p seq(print(seq(round(evalf(T(n,k),32)),k=0..n)),n=0..7);

%p # _Peter Luschny_, Aug 16 2012

%t T[n_, n_] := CatalanNumber[n]; T[n_, k_] /; 0 <= k < n := T[n-1, k] + T[n, k+1]; T[_, _] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* _Jean-François Alcover_, Jun 11 2019 *)

%o (Sage)

%o def T(n, k) :

%o if k > n : return 0

%o if n == k : return binomial(2*n, n)/(n+1)

%o return T(n-1, k) + T(n, k+1)

%o A106534 = lambda n,k: T(n, k)

%o for n in (0..5): [A106534(n,k) for k in (0..n)] # _Peter Luschny_, Aug 16 2012

%o (Magma)

%o function T(n,k)

%o if k gt n then return 0;

%o elif k eq n then return Catalan(n);

%o else return T(n-1, k) + T(n, k+1);

%o end if; return T;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 18 2021

%Y Columns: A007317, A002212, see also A045868, A055452-A055455.

%Y Diagonals: A000108, A005807.

%Y Cf. A059346 (Catalan difference array as triangle).

%K nonn,easy,tabl

%O 0,2

%A _Philippe Deléham_, May 30 2005