# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a160563 Showing 1-1 of 1 %I A160563 #24 Mar 03 2024 14:13:09 %S A160563 1,1,1,9,10,1,225,259,35,1,11025,12916,1974,84,1,893025,1057221, %T A160563 172810,8778,165,1,108056025,128816766,21967231,1234948,28743,286,1, %U A160563 18261468225,21878089479,3841278805,230673443,6092515,77077,455,1,4108830350625,4940831601000 %N A160563 Table of the number of (n,k)-Riordan complexes, read by rows. %C A160563 From Table 4, right-hand side, of Gelineau and Zeng. %C A160563 Essentially a row-reversal of A008956. - _R. J. Mathar_, May 20 2009 %H A160563 Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 2009. %H A160563 W. Zhang, Some identities involving the Euler and the central factorial numbers, The Fibonacci Quarterly, Vol. 36, Number 2, May 1998. %F A160563 a(n,k) = |v(n,k)| where v(n,k) = v(n-1,k-1) - (2n-1)^2*v(n-1,k); eq (4.2). %F A160563 Let F(x) = 1/cos(x). Then (2*n)!*(1/cos(x))^(2*n+1) = Sum_{k=0..n} T(n,k)*F^(2*k)(x), where F^(r) denotes the r-th derivative of F(x) (Zhang 1998). An example is given below. - _Peter Bala_, Feb 06 2012 %F A160563 Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=1..n step 2, n=1..len step 2] the 'odd subtriangle' of U. This triangle is the odd subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit with signed terms. See A182867 for the even subtriangle. - _Peter Luschny_, Mar 03 2024 %e A160563 Triangle starts: %e A160563 [0] 1; %e A160563 [1] 1, 1; %e A160563 [2] 9, 10, 1; %e A160563 [3] 225, 259, 35, 1; %e A160563 [4] 11025, 12916, 1974, 84, 1; %e A160563 [5] 893025, 1057221, 172810, 8778, 165, 1; %e A160563 [6] 108056025, 128816766, 21967231, 1234948, 28743, 286, 1; %e A160563 . %e A160563 For row 3: F(x) := 1/cos(x). Then 225*F(x) + 259*(d/dx)^2(F(x)) + 35*(d/dx)^4(F(x)) + (d/dx)^6(F(x)) = 720*(1/cos(x))^7, where F^(r) denotes the r-th derivative of F(x). %p A160563 t := proc(n,k) option remember ; expand(x*mul(x+n/2-i,i=1..n-1)) ; coeftayl(%,x=0,k) ; end: %p A160563 v := proc(n,k) option remember ; 4^(n-k)*t(2*n+1,2*k+1) ; end: %p A160563 A160563 := proc(n,k) abs(v(n,k)) ; end: for n from 0 to 10 do for k from 0 to n do printf("%d,",A160563(n,k)) ; od: od: # _R. J. Mathar_, May 20 2009 %p A160563 # Using a bivariate generating function (albeit generating signed terms): %p A160563 gf := (t + sqrt(1 + t^2))^x: ser := series(gf, t, 20): %p A160563 ct := n -> coeff(ser, t, n): T := (n, k) -> n!*coeff(ct(n), x, k): %p A160563 OddPart := (T, len) -> local n, k; %p A160563 seq(print(seq(T(n, k), k = 1..n, 2)), n = 1..2*len, 2): %p A160563 OddPart(T, 6); # _Peter Luschny_, Mar 03 2024 %t A160563 t[_, 0] = 1; t[n_, n_] := t[n, n] = ((2*n - 1)!!)^2; t[n_, k_] := t[n, k] = (2*n - 1)^2*t[n - 1, k - 1] + t[n - 1, k]; %t A160563 T[n_, k_] := t[n, n - k]; %t A160563 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 28 2017, after _R. J. Mathar_'s comment *) %Y A160563 Cf. A001819, A008275, A008277, A160562, A091885, A182867. %K A160563 nonn,tabl %O A160563 0,4 %A A160563 _Jonathan Vos Post_, May 19 2009 %E A160563 Extended by _R. J. Mathar_, May 20 2009 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE