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 A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n. 32
 1, 1, 6, 1, 30, 90, 1, 126, 1260, 2520, 1, 510, 13230, 75600, 113400, 1, 2046, 126720, 1580040, 6237000, 7484400, 1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400, 1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000, 1, 131070, 96461910, 8203431600, 196556560200, 1882311631200, 8266953895200, 16672848192000, 12504636144000 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS T(n,k) gives the number of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. An example is given below. Cf. A019538 and A156289. - Peter Bala, Aug 20 2014 REFERENCES H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283. S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126. S. A. Joffe, Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 48 (1917-1920), 193-271. LINKS Muniru A Asiru, Rows n = 1..50 of triangle, flattened FORMULA T(n,k) = 0 if k <= 0 or k > n, = 1 if k=1, otherwise T(n,k) = k*(2*k-1)*T(n-1,k-1) + k^2*T(n-1,k). Related to Euler numbers A000364 by A000364(n) = (-1)^n*Sum_{k=1..n} (-1)^k*T(n,k). For example, A000364(3) = 61 = 90 - 30 + 1. From Peter Bala, Aug 20 2014: (Start) T(n,k) = 1/(2^(k-1))*Sum_{j = 1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n). T(n,k) = k!*A156289(n,k) = k!*(2*k-1)!!*A036969. E.g.f.: A(t,z) := 1/( 1 - t*(cosh(z) - 1) ) = 1 + t*z^2/2! + (t + 6*t^2)*z^4/4! + (t + 30*t^2 + 90*t^3)*z^6/6! + ... satisfies the partial differential equation d^2/dz^2(A) = D(A), where D = t^2*(2*t + 1)*d^2/dt^2 + t*(5*t + 1)*d/dt + t. Hence the row polynomials R(n,t) satisfy the differential equation R(n+1,t) = t^2*(2*t + 1)*R''(n,t) + t*(5*t + 1)*R'(n,t) + t*R(n,t) with R(0,t) = 1, where ' indicates differentiation w.r.t. t. This is equivalent to the above recurrence equation. Recurrence for row polynomials: R(n,t) = t*( Sum_{k = 1..n} binomial(2*n,2*k)*R(n-k,t) ) with R(0,t) := 1. Row sums equal A094088(n) for n >= 1. A100872(n) = (1/2)*R(n,2). (End) EXAMPLE Triangle begins: 1, 1, 6, 1, 30, 90, 1, 126, 1260, 2520, 1, 510, 13230, 75600, 113400, 1, 2046, 126720, 1580040, 6237000, 7484400, 1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400, 1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000, ... From Peter Bala, Aug 20 2014: (Start) Row 2: [1,6] k Ordered set partitions of {1,2,3,4} into k blocks Number - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 {1,2,3,4} 1 2 {1,2}{3,4}, {3,4}{1,2}, {1,3}{2,4}, {2,4}{1,3}, 6 {1,4}{2,3}, {2,3}{1,4} (End) MAPLE T := proc(n, k) option remember; if k > n then 0 elif k=0 then k^n elif k=1 then 1 else k*(2*k-1)*T(n-1, k-1)+k^2*T(n-1, k); fi; end: # Minor edit to make it also work in the (0, 0)-offset case. Peter Luschny, Sep 03 2022 for n from 1 to 12 do lprint([seq(T(n, k), k=1..n)]); od: MATHEMATICA T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[_, 1] = 1; T[_, _] = 0; Table[T[n, k], {n, 1, 9}, {k, 1, n}] (* Jean-François Alcover, Jul 03 2019 *) PROG (Sage) @cached_function def A241171(n, k): if n == 0 and k == 0: return 1 if k < 0 or k > n: return 0 return (2*k^2 - k)*A241171(n - 1, k - 1) + k^2*A241171(n - 1, k) for n in (1..6): print([A241171(n, k) for k in (1..n)]) # Peter Luschny, Sep 06 2017 (GAP) Flat(List([1..10], n->List([1..n], k->1/(2^(k-1))*Sum([1..k], j->(-1)^(k-j)*Binomial(2*k, k-j)*j^(2*n))))); # Muniru A Asiru, Feb 27 2019 CROSSREFS Case m=2 of the polynomials defined in A278073. Cf. A000680 (diagonal), A094088 (row sums), A000364 (alternating row sums), A281478 (central terms), A327022 (refinement). Diagonals give A002446, A213455, A241172, A002456. Cf. A019538, A036969, A156289. Sequence in context: A178726 A030524 A327022 * A051930 A347488 A147320 Adjacent sequences: A241168 A241169 A241170 * A241172 A241173 A241174 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Apr 22 2014 STATUS approved

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Last modified February 21 09:29 EST 2024. Contains 370228 sequences. (Running on oeis4.)