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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]

[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: A318607 A259824 A065173 * 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 October 20 15:29 EDT 2019. Contains 328267 sequences. (Running on oeis4.)