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A103364 Matrix inverse of the Narayana triangle A001263. 8
1, -1, 1, 2, -3, 1, -7, 12, -6, 1, 39, -70, 40, -10, 1, -321, 585, -350, 100, -15, 1, 3681, -6741, 4095, -1225, 210, -21, 1, -56197, 103068, -62916, 19110, -3430, 392, -28, 1, 1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1, -27036487, 49615695, -30346380, 9276120, -1698732, 206388 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The first column is A103365. The second column is A103366. Row sums are all zeros (for n>1). Absolute row sums form A103367.

Let E(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = 1/sqrt(y)*BesselI(1,2*sqrt(y)). Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang. - Peter Bala, Aug 07 2013

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

FORMULA

From Peter Bala, Aug 07 2013: (Start)

Let E(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = 1/sqrt(y)* BesselI(1,2*sqrt(y)). Generating function: E(x*y)/E(y) = 1 + (-1 + x)*y/(1!*2!) + (2 - 3*x + x^2)*y^2/(2!*3!) + (-7 + 12*x - 6*x^2 + x^3)*y^3/(3!*4!) + .... The n-th power of this array has a generating function E(x*y)/E(y)^n. In particular, the matrix inverse A001263 has a generating function E(y)*E(x*y).

Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} 1/(n-k+1)*binomial(n,k)*binomial(n+1,k+1) *R(k,x) with initial value R(0,x) = 1.

Let alpha denote the root of E(x) = 0 that is smallest in absolute magnitude. Numerically, alpha = - 3.67049 26605 ... = -(A115369/2)^2. It appears that for arbitrary complex x we have lim {n -> inf} R(n,x)/R(n,0) = E(alpha*x). Cf. A055133, A086646 and A104033.

A stronger result than pointwise convergence may hold: the convergence may be uniform on compact subsets of the complex plane. This would explain the observation that the real zeros of the polynomials R(n,x) seem to converge to the zeros of E(alpha*x) as n increases. Some numerical examples are given below. (End)

From Werner Schulte, Jan 04 2017: (Start)

T(n,k) = T(n-1,k-1)*n*(n-1)/(k*(k-1)) for 1 < k <= n;

T(n,k) = T(n+1-k,1)*A001263(n,k) for 1 <= k <= n;

Sum_{k=1..n} T(n,k)*A000108(n) = 1 for n > 0. (End)

EXAMPLE

Rows begin:

        1;

       -1,        1;

        2,       -3,       1;

       -7,       12,      -6,       1;

       39,      -70,      40,     -10,     1;

     -321,      585,    -350,     100,   -15,     1;

     3681,    -6741,    4095,   -1225,   210,   -21,   1;

   -56197,   103068,  -62916,   19110, -3430,   392, -28,   1;

  1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1;

  ...

From Peter Bala, Aug 09 2013: (Start)

The real zeros of the row polynomials R(n,x) appear to converge to zeros of E(alpha*x) as n increases, where alpha = - 3.67049 26605 ... ( = -(A115369/2)^2).

Polynomial | Real zeros to 5 decimal places

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

R(5,x)     | 1, 3.57754, 3.81904

R(10,x)    | 1, 3.35230, 7.07532,  9.14395

R(15,x)    | 1, 3.35231, 7.04943, 12.09668, 15.96334

R(20,x)    | 1, 3.35231, 7.04943, 12.09107, 18.47845, 24.35255

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Function   |

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

E(alpha*x) | 1, 3.35231, 7.04943, 12.09107, 18.47720, 26.20778, ...

Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( BesselJZeros(1,n)/ BesselJZeros(1,1))^2 ). (End)

MATHEMATICA

T[n_, 1]:= Last[Table[(-1)^(n - 1)*(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 1000}], x])[[k]]*(n)!*(n - 1)!, {k, 1, 2*n - 1, 2}]]

T[n_, n_] := 1; T[2, 1] := -1; T[3, 1] := 2; T[n_, k_] := T[n, k] = T[n - 1, k - 1]*n*(n - 1)/(k*(k - 1)); Table[T[n, k], {n, 1, 50}, {k, 1, n}] // Flatten (* G. C. Greubel, Jan 04 2016 *)

PROG

(PARI) T(n, k)=if(n<k || k<1, 0, (matrix(n, n, m, j, binomial(m-1, j-1)*binomial(m, j-1)/j)^-1)[n, k])

CROSSREFS

Cf. A000108, A001263, A055133, A086646, A103365, A103366, A103367, A104033.

Sequence in context: A101175 A050512 A107102 * A104027 A192363 A097710

Adjacent sequences:  A103361 A103362 A103363 * A103365 A103366 A103367

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Feb 02 2005

STATUS

approved

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Last modified February 24 07:50 EST 2018. Contains 299599 sequences. (Running on oeis4.)