This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103364 #44 May 05 2025 18:32:50
%S A103364 1,-1,1,2,-3,1,-7,12,-6,1,39,-70,40,-10,1,-321,585,-350,100,-15,1,
%T A103364 3681,-6741,4095,-1225,210,-21,1,-56197,103068,-62916,19110,-3430,392,
%U A103364 -28,1,1102571,-2023092,1236816,-377496,68796,-8232,672,-36,1,-27036487,49615695,-30346380,9276120,-1698732,206388
%N A103364 Matrix inverse of the Narayana triangle A001263.
%C A103364 The first column is A103365. The second column is A103366. Row sums are all zeros (for n > 1). Absolute row sums form A103367.
%C A103364 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
%H A103364 G. C. Greubel, <a href="/A103364/b103364.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A103364 W. Wang and T. Wang, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
%F A103364 From _Peter Bala_, Aug 07 2013: (Start)
%F A103364 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).
%F A103364 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.
%F A103364 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->oo} R(n,x)/R(n,0) = E(alpha*x). Cf. A055133, A086646 and A104033.
%F A103364 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)
%F A103364 From _Werner Schulte_, Jan 04 2017, corrected May 05 2025: (Start)
%F A103364 T(n,k) = T(n-1,k-1)*n*(n-1)/(k*(k-1)) for 1 < k <= n;
%F A103364 T(n,k) = T(n+1-k,1)*A001263(n,k) for 1 <= k <= n;
%F A103364 Sum_{k=1..n} T(n,k)*A000108(k) = 1 for n > 0. (End)
%e A103364 Rows begin:
%e A103364 1;
%e A103364 -1, 1;
%e A103364 2, -3, 1;
%e A103364 -7, 12, -6, 1;
%e A103364 39, -70, 40, -10, 1;
%e A103364 -321, 585, -350, 100, -15, 1;
%e A103364 3681, -6741, 4095, -1225, 210, -21, 1;
%e A103364 -56197, 103068, -62916, 19110, -3430, 392, -28, 1;
%e A103364 1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1;
%e A103364 ...
%e A103364 From _Peter Bala_, Aug 09 2013: (Start)
%e A103364 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).
%e A103364 Polynomial | Real zeros to 5 decimal places
%e A103364 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A103364 R(5,x) | 1, 3.57754, 3.81904
%e A103364 R(10,x) | 1, 3.35230, 7.07532, 9.14395
%e A103364 R(15,x) | 1, 3.35231, 7.04943, 12.09668, 15.96334
%e A103364 R(20,x) | 1, 3.35231, 7.04943, 12.09107, 18.47845, 24.35255
%e A103364 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A103364 Function |
%e A103364 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A103364 E(alpha*x) | 1, 3.35231, 7.04943, 12.09107, 18.47720, 26.20778, ...
%e A103364 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)
%t A103364 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 A103364 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 *)
%t A103364 T[n_, n_] := 1; T[n_, 1]/;n>1 := T[n, 1] = -Sum[T[n, j], {j, 2, n}]; T[n_, k_]/;1<k<n := T[n, k] = T[n-1, k-1] n(n-1)/k/(k-1); T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* _Oliver Seipel_, Jan 01 2025 *)
%o A103364 (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])
%Y A103364 Cf. A000108, A001263, A055133, A086646, A103365, A103366, A103367, A104033.
%K A103364 sign,tabl
%O A103364 1,4
%A A103364 _Paul D. Hanna_, Feb 02 2005