A124644 - OEIS

%I #50 Feb 04 2024 18:22:53

%S 1,1,1,2,2,1,5,6,3,1,14,20,12,4,1,42,70,50,20,5,1,132,252,210,100,30,

%T 6,1,429,924,882,490,175,42,7,1,1430,3432,3696,2352,980,280,56,8,1,

%U 4862,12870,15444,11088,5292,1764,420,72,9,1,16796,48620,64350,51480,27720

%N Triangle read by rows. T(n, k) = binomial(n, k) * CatalanNumber(n - k).

%C Equal to A091867*A007318. - _Philippe Deléham_, Dec 12 2009

%C Exponential Riordan array [exp(2x)*(Bessel_I(0,2x)-Bessel_I(1,2x)),x]. - _Paul Barry_, Mar 03 2011

%C From _Tom Copeland_, Nov 04 2014: (Start)

%C O.g.f: G(x,t) = C[Pinv(x,t)] =  {1 - sqrt[1 - 4 *x /(1-x*t)]}/2 where C(x) = [1 - sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108 with inverse Cinv(x) = x*(1-x), and Pinv(x,t)= -P(-x,t) = x/(1-t*x) with inverse P(x,t) = 1/(1+t*x). This puts this array in a family of arrays formed from the composition of C and P and their inverses. -G(-x,t) is the comp. inverse of the o.g.f. of A030528.

%C This is an Appell sequence with lowering operator d/dt p(n,t) = n*p(n-1,t) and (p(.,t)+a)^n = p(n,t+a). The e.g.f. has the form e^(x*t)/w(t) where 1/w(t) is the e.g.f. of the first column, which is the Catalan sequence A000108. (End)

%H Indranil Ghosh, <a href="/A124644/b124644.txt">Rows 0..125, flattened</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6.

%F T(n,k) = [x^(n-k)]F(-n,n-k+1;1;-1-x). - _Paul Barry_, Sep 05 2008

%F G.f.: 1/(1-xy-x/(1-x/(1-xy-x/(1-x/(1-xy-x/(1-x.... (continued fraction). - _Paul Barry_, Jan 06 2009

%F G.f.: 1/(1-x-xy-x^2/(1-2x-xy-x^2/(1-2x-xy-x^2/(1-.... (continued fraction). - _Paul Barry_, Jan 28 2009

%F T(n,k) = Sum_{i = 0..n} C(n,i)*(-1)^(n-i)*Sum{j = 0..i} C(j,k)*C(i,j)*A000108(i-j). - _Paul Barry_, Aug 03 2009

%F Sum_{k = 0..n} T(n,k)*x^k = A126930(n), A005043(n), A000108(n), A007317(n+1), A064613(n), A104455(n) for x = -2, -1, 0, 1, 2, 3 respectively. T(n,k)= A007318(n,k)*A000108(n-k). - _Philippe Deléham_, Dec 12 2009

%F E.g.f.: exp(2*x + x*y)*(Bessel_I(0,2*x) - Bessel_I(1,2*x)). - _Paul Barry_, Mar 10 2010

%F From _Tom Copeland_, Nov 08 2014: (Start)

%F O.g.f.: G(x,t) = C[P(x,t)] = [1 - sqrt(1-4*x / (1-t*x))] / 2 = Sum_{n >= 1} (C. +  t)^(n-1) * x^n] = x + (1 + t) x^2 + (2 + 2t + t^2) x^3 + ... umbrally, where (C.)^n = C_n = (1,1,2,5,8,...) = A000108(x), C(x)= x*A000108(x)= G(x,0), and P(x,t) = x/(1 + t*x), a special linear fractional (Mobius) transformation. P(x,-t)= -P(-x,t) is the inverse of P(x,t).

%F Inverse o.g.f.: Ginv(x,t) = P[Cinv(x),-t] = x*(1-x) / [1 - t*x(1-x)] = -A030528(-x,t), where Cinv(x) = x*(1-x) is the inverse of C(x).

%F G(x,t) = x*A091867(x,t+1), and Ginv(x,t) = x*A104597(x,-(t+1)). (End)

%F T(n, k) = (-1)^(n-k)*Catalan(n-k)*Pochhammer(-n,n-k)/(n-k)!. - _Peter Luschny_, Feb 05 2015

%F Recurrence: T(n, 0) = Catalan(n) = 1/(n+1)*binomial(2*n, n) and, for 1 <= k <= n, T(n, k) = (n/k) * T(n-1, k-1). - _Peter Bala_, Feb 04 2024

%e From _Paul Barry_, Jan 28 2009: (Start)

%e Triangle begins

%e    1,

%e    1,  1,

%e    2,  2,  1,

%e    5,  6,  3,  1,

%e   14, 20, 12,  4,  1,

%e   42, 70, 50, 20,  5,  1 (End)

%p m:=n->binomial(2*n, n)/(n+1): T:=proc(n, k) if k<=n then binomial(n, k)*m(n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od;

%t Table[Binomial[n, #] Binomial[2 #, #]/(# + 1) &[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* or *)

%t Table[Abs[(-1)^k*CatalanNumber[#] Pochhammer[-n, #]/#!] &[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Feb 17 2017 *)

%o (Sage)

%o def A124644(n,k):

%o     return (-1)^(n-k)*catalan_number(n-k)*rising_factorial(-n,n-k)/factorial(n-k)

%o for n in range(7): [A124644(n,k) for k in (0..n)] # _Peter Luschny_, Feb 05 2015

%Y Cf. A098474 (mirror image), A000108, A091867, A030528, A104597.

%Y Row sums give A007317(n+1).

%K nonn,tabl,easy

%O 0,4

%A Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006

%E Name brought in line with the Maple program by _Peter Luschny_, Jun 21 2023