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 A098474 Triangle read by rows, T(n,k) = C(n,k)*C(2*k,k)/(k+1), n>=0, 0<=k<=n. 11
 1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 20, 14, 1, 5, 20, 50, 70, 42, 1, 6, 30, 100, 210, 252, 132, 1, 7, 42, 175, 490, 882, 924, 429, 1, 8, 56, 280, 980, 2352, 3696, 3432, 1430, 1, 9, 72, 420, 1764, 5292, 11088, 15444, 12870, 4862, 1, 10, 90, 600, 2940, 10584, 27720 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS A Catalan scaled binomial matrix. From Philippe Deléham, Sep 01 2005: (Start) Table U(n,k),k>=0, n>=0, read by antidiagonals, begins: row k = 0 : 1, 1, 2, 5, 14, 42, ...  is A000108 row k = 1 : 1, 2, 6, 20, 70, ...     is A000984 row k = 2 : 1, 3, 12, 50, 280, ...   is A007854 row k = 3 : 1, 4, 20, 104, 548, ...  is A076035 row k = 4 : 1, 5, 30, 185, 1150, ... is A076036 G.f. for row k : 1/(1-(k+1)*x*C(x)) where C(x) is the g.f. = for Catalan numbers A000108. U(n,k) = sum_{j, 0<=j<=n} A106566(n,j)*(k+1)^j. (End) This sequence gives the coefficients (increasing powers of x) of the Jensen polynomials for the Catalan sequence A000108 of degree n and shift 0. For the definition of Jensen polynomials for a sequence see a comment in A094436. - Wolfdieter Lang, Jun 25 2019 LINKS Indranil Ghosh, Rows 0..125, flattened FORMULA G.f.: 2/(1-x+(1-x-4*x*y)^(1/2)). - Vladeta Jovovic, Sep 11 2004 E.g.f.: exp(x*(1+2*y))*(BesselI(0, 2*x*y)-BesselI(1, 2*x*y)). - Vladeta Jovovic, Sep 11 2004 G.f.: 1/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-x-xy/(1-xy/(1-... (continued fraction). - Paul Barry, Feb 11 2009 Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A126930(n), A005043(n), A000108(n), A007317(n+1), A064613(n), A104455(n) for x = -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Dec 12 2009 T(n,k) = (-1)^k*Catalan(k)*Pochhammer(-n,k)/k!. - Peter Luschny, Feb 05 2015 O.g.f.: [1 - sqrt(1-4tx/(1-x))]/(2tx) = 1 + (1+t) x + (1+2t+2t^2) x^2 + (1+3t+6t^2+5t^3) x^3 + ... , generating the polynomials of this entry, reverse of A124644. See A011973 for a derivation and the inverse o.g.f., connected to the Fibonacci, Chebyshev, and Motzkin polynomials. See also A267633. - Tom Copeland, Jan 25 2016 From Peter Bala, Jun 13 2016: (Start) The o.g.f. F(x,t) = ( 1 - sqrt(1 - 4*t*x/(1 - x)) )/(2*t*x) satisfies the partial differential equation d/dx(x*(1 - x)*F) - x*t*(1 + 4*t)*dF/dt - 2*x*t*F = 1. This gives a recurrence for the row polynomials: (n + 2)*R(n+1,t) = t*(1 + 4*t)*R'(n,t) + (2*t + n + 2)*R(n,t), where the prime ' indicates differentiation with respect to t. Equivalently, setting Q(n,t) = t^(n+2)*R(n,-t)/(1 - 4*t)^(n + 3/2) we have t^2*d/dt(Q(n,t)) = (n + 2)*Q(n+1,t). This leads to the following expansions: Q(0,t) = 1/2*Sum_{k >= 1} k*binomial(2*k,k)*t^(k+1) Q(1,t) = 1/2*Sum_{k >= 1} k*(k+1)/2!*binomial(2*k,k)*t^(k+2) Q(2,t) = 1/2*Sum_{k >= 1} k*(k+1)*(k+2)/3!*binomial(2*k,k) *t^(k+3) and so on. (End) EXAMPLE Rows begin:  [1, 1] [1, 2,  2] [1, 3,  6,   5] [1, 4, 12,  20,  14] [1, 5, 20,  50,  70,  42] [1, 6, 30, 100, 210, 252, 132] ... Row 3: t*(1 - 3*t + 6*t^2 - 5*t^3)/(1 - 4*t)^(9/2) = 1/2*Sum_{k >= 1} k*(k+1)*(k+2)*(k+3)/4!*binomial(2*k,k)*t^k. - Peter Bala, Jun 13 2016 MATHEMATICA Table[Binomial[n, k] Binomial[2 k, k]/(k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* or *) Table[(-1)^k*CatalanNumber[k] Pochhammer[-n, k]/k!, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 17 2017 *) PROG (Sage) def A098474(n, k):     return (-1)^k*catalan_number(k)*rising_factorial(-n, k)/factorial(k) for n in range(7): [A098474(n, k) for k in (0..n)] # Peter Luschny, Feb 05 2015 CROSSREFS Row sums are A007317. Antidiagonal sums are A090344. Principal diagonal is A000108. Mirror image of A124644. Cf. A000984, A007854, A011973, A076035, A076036, A09443, A098473, A106566, A267633. Sequence in context: A259824 A065173 A330965 * A153199 A056860 A158825 Adjacent sequences:  A098471 A098472 A098473 * A098475 A098476 A098477 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Sep 09 2004 EXTENSIONS New name using a formula of Paul Barry by Peter Luschny, Feb 05 2015 STATUS approved

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Last modified January 19 14:53 EST 2020. Contains 331049 sequences. (Running on oeis4.)