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A185411 A triangular decomposition of the double factorial numbers A001147. 6
1, 0, 1, 0, 2, 1, 0, 4, 10, 1, 0, 8, 60, 36, 1, 0, 16, 296, 516, 116, 1, 0, 32, 1328, 5168, 3508, 358, 1, 0, 64, 5664, 42960, 64240, 21120, 1086, 1, 0, 128, 23488, 320064, 900560, 660880, 118632, 3272, 1, 0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are A001147. Reversal of A185410. Contains A156919 as submatrix.

Row n counts perfect matchings of [2n] by number of matches in which the smaller entry is odd. For example, T(2,1)=2 counts 13/24, 14/23, in each of which only the first matching pair has an odd smaller entry. Outline proof. Consider the map on perfect matchings of [2n] given by "delete the entries n and n-1 and, if they were not originally matched to each other, match up their now-unmatched partners". Consideration of this map and its effect on the statistic "number of matches in which the smaller entry is odd" yields the Mathematica recurrence below. - David Callan, Dec 13 2011

Triangle T(n,k), 0<=k<=n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 11, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 12 2013

LINKS

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

Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv: 1204.4963v3 [math.CO], 2012.

Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, Elect. J. Combinat. 20 (1) (2013) #P11.

Shi-Mei Ma, T. Mansour, D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.

S.-M. Ma, T. Mansour and D. G. L. Wang, Combinatorics of Dumont differential system on the Jacobi elliptic functions, arXiv:1403.0233 [math.CO], 2014

S.-M. Ma, Y.-N. Yeh, Stirling permutations, cycle structures of permutations and perfect matchings, arXiv:1503.06601 [math.CO], 2015.

FORMULA

G.f.: 1/(1-xy/(1-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1- ... (continued fraction).

T(n,k) = (2n-2k+1)*T(n-1,k-1) + 2k*T(n-1,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 12 2013

T(n,k) = 2^(n-k)*A211399(n,k). - Philippe Deléham, Feb 12 2013

EXAMPLE

1,

0,   1,

0,   2,     1,

0,   4,    10,       1,

0,   8,    60,      36,        1,

0,  16,   296,     516,      116,        1,

0,  32,  1328,    5168,     3508,      358,       1,

0,  64,  5664,   42960,    64240,    21120,    1086,      1,

0, 128, 23488,  320064,   900560,   660880,  118632,   3272,    1,

0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1

MATHEMATICA

u[n_, 0] := If[n==0, 1, 0]; 
u[n_, m_] /; m==1 := 2^(n - 1); 
u[n_, m_] /; m==n>=1 := 1; 
u[n_, m_] /; 1<m<n := u[n, m] = (2m)*u[n - 1, m] + (2n - 2m + 1)*u[n - 1, m - 1]; Flatten[
Table[u[n, m], {n, 0, 10}, {m, 0, n}]] (* David Callan, Dec 13 2011 *)

CROSSREFS

Columns 0-1 give: A000007, A131577.

Cf. A001147, A185410, A156919 (another version).

Sequence in context: A160168 A077929 A178039 * A254882 A086095 A112334

Adjacent sequences:  A185408 A185409 A185410 * A185412 A185413 A185414

KEYWORD

nonn,easy,tabl

AUTHOR

Paul Barry, Jan 26 2011

EXTENSIONS

Sequence terms corrected by Paul Barry, Jan 27 2011

STATUS

approved

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Last modified February 21 12:27 EST 2018. Contains 299411 sequences. (Running on oeis4.)