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%I #41 Aug 31 2023 11:02:15
%S 1,0,1,0,0,1,0,1,0,1,0,0,4,0,1,0,1,0,10,0,1,0,0,16,0,20,0,1,0,1,0,91,
%T 0,35,0,1,0,0,64,0,336,0,56,0,1,0,1,0,820,0,966,0,84,0,1,0,0,256,0,
%U 5440,0,2352,0,120,0,1,0,1,0,7381,0,24970,0,5082,0,165,0,1,0,0,1024,0,87296,0
%N Triangular array: T(n,k) counts the partitions of the set [n] into k odd sized blocks.
%C For partitions into blocks of even size see A156289.
%C Essentially the unsigned matrix inverse of triangle A121408.
%C From _Peter Bala_, Jul 28 2014: (Start)
%C Define a polynomial sequence x_(n) by setting x_(0) = 1 and for n = 1,2,... setting x_(n) = x*(x + n - 2)*(x + n - 4)*...*(x + n - 2*(n - 1)). Then this table is the triangle of connection constants for expressing the monomial polynomials x^n in terms of the basis x_(k), that is, x^n = sum {k = 0..n} T(n,k)*x_(k) for n = 0,1,2,.... An example is given below.
%C Let M denote the lower unit triangular array A119467 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
%C /I_k 0\
%C \ 0 M/ having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle, omitting the first row and column, equals the infinite matrix product M(0)*M(1)*M(2)*.... (End)
%C Also the Bell transform of A000035(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%D L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 225-226.
%H Ch. A. Charalambides, <a href="http://www.fq.math.ca/Scanned/19-5/charalambides.pdf">Central factorial numbers and related expansions</a>, Fib. Quarterly, Vol. 19, No 5, Dec 1981, pp. 451-456.
%H Feng Qi and Peter Taylor, <a href="https://www.researchgate.net/profile/Feng-Qi-26/publication/373421001_SERIES_EXPANSIONS_FOR_POWERS_OF_SINC_FUNCTION_AND_CLOSED-FORM_EXPRESSIONS_FOR_SPECIFIC_PARTIAL_BELL_POLYNOMIALS/">Series expansions for powers of sinc function and closed-form expressions for specific partial Bell polynomials</a>, Appl. Anal. Disc. Math. (2024) Vol. 18, No. 1, 1-24. See p. 13.
%F G.f. for column k: x^k/Product_{j=0..floor(k/2)} (1 - (2*j + k-2*floor(k/2))^2 * x^2).
%F G.f. for column 2*k: x^(2*k)/Product_{j=0..k} (1 - (2*j)^2*x^2).
%F G.f. for column 2*k+1: x^(2*k+1)/Product_{j=0..k} (1 - (2*j+1)^2*x^2).
%F From _Peter Bala_, Feb 21 2011 (Start)
%F T(n,k) = 1/(2^k*k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j-k)^n,
%F Recurrence relation T(n+2,k) = T(n,k-2) + k^2*T(n,k).
%F E.g.f.: F(x,z) = exp(x*sinh(z)) = Sum_{n>=0} R(n,x)*z^n/n! = 1 + x*z + x^2*z^2/2! + (x+x^3)*z^3/3! + ....
%F The row polynomials R(n,x) begin
%F R(1,x) = x
%F R(2,x) = x^2
%F R(3,x) = x+x^3.
%F The e.g.f. F(x,z) satisfies the partial differential equation d^2/dz^2(F) = x^2*F + x*F' + x^2*F'' where ' denotes differentiation w.r.t. x.
%F Hence the row polynomials satisfy the recurrence relation R(n+2,x) = x^2*R(n,x) + x*R'(n,x) + x^2*R''(n,x) with R(0,x) = 1.
%F The recurrence relation for T(n,k) given above follows from this.
%F (End)
%F For the corresponding triangle of ordered partitions into odd-sized blocks see A196776. Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of exp(t*M) lists the row polynomials for the present triangle. - _Peter Bala_, Oct 06 2011
%F Row generating polynomials equal D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A196776. - _Peter Bala_, Dec 06 2011
%F From _Peter Bala_, Jul 28 2014: (Start)
%F E.g.f.: exp(t*sinh(x)) = 1 + t*x + t^2*x^2/2! + (t + t^3)*x^3/3! + ....
%F Hockey-stick recurrence: T(n+1,k+1) = Sum_{i = 0..floor((n-k)/2)} binomial(n,2*i)*T(n-2*i,k).
%F Recurrence equation for the row polynomials R(n,t):
%F R(n+1,t) = t*Sum_{k = 0..floor(n/2)} binomial(n,2*k)*R(n-2*k,t) with R(0,t) = 1. (End)
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 0, 1;
%e 0, 1, 0, 1;
%e 0, 0, 4, 0, 1;
%e 0, 1, 0, 10, 0, 1;
%e 0, 0, 16, 0, 20, 0, 1;
%e 0, 1, 0, 91, 0, 35, 0, 1;
%e 0, 0, 64, 0, 336, 0, 56, 0, 1;
%e 0, 1, 0, 820, 0, 966, 0, 84, 0, 1;
%e 0, 0, 256, 0, 5440, 0, 2352, 0, 120, 0, 1;
%e 0, 1, 0, 7381, 0, 24970, 0, 5082, 0, 165, 0, 1;
%e T(5,3) = 10. The ten partitions of the set [5] into 3 odd-sized blocks are
%e (1)(2)(345), (1)(3)(245), (1)(4)(235), (1)(5)(234), (2)(3)(145),
%e (2)(4)(135), (2)(5)(134), (3)(4)(125), (3)(5)(124), (4)(5)(123).
%e Connection constants: Row 5 = [0,1,0,10,0,1]. Hence, with the polynomial sequence x_(n) as defined in the Comments section we have x^5 = x_(1) + 10*x_(3) + x_(5) = x + 10*x*(x+1)*(x-1) + x*(x+3)*(x+1)*(x-1)*(x-3).
%p A136630 := proc (n, k) option remember; if k < 0 or n < k then 0 elif k = n then 1 else procname(n-2, k-2) + k^2*procname(n-2, k) end if end proc: seq(seq(A136630(n, k), k = 1 .. n), n = 1 .. 12); # _Peter Bala_, Jul 27 2014
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> (n+1) mod 2, 9); # _Peter Luschny_, Jan 27 2016
%t t[n_, k_] := Coefficient[ x^k/Product[ 1 - (2*j + k - 2*Quotient[k, 2])^2*x^2, {j, 0, k/2}] + x*O[x]^n, x, n]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 22 2013, after Pari *)
%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
%t rows = 13;
%t M = BellMatrix[Mod[#+1, 2]&, rows];
%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%o (PARI) {T(n,k)=polcoeff(x^k/prod(j=0,k\2,1-(2*j+k-2*(k\2))^2*x^2 +x*O(x^n)),n)}
%Y Cf. A121408; A136631 (antidiagonal sums), A003724 (row sums), A136632; A002452 (column 3), A002453 (column 5); A008958 (central factorial triangle), A156289. A185690, A196776.
%K nonn,tabl
%O 0,13
%A _Paul D. Hanna_, Jan 14 2008
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