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%I #62 Oct 05 2023 15:19:59
%S 1,0,1,1,1,1,2,3,2,1,9,11,7,3,1,44,53,32,13,4,1,265,309,181,71,21,5,1,
%T 1854,2119,1214,465,134,31,6,1,14833,16687,9403,3539,1001,227,43,7,1,
%U 133496,148329,82508,30637,8544,1909,356,57,8,1
%N Triangle T(n, k), read by row, related to Euler's difference table A068106 (divide column k of A068106 by k!).
%C The k-th column sequence, k >= 0, without leading zeros, enumerates the ways to distribute n beads, n >= 1, labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and k+1 indistinguishable, ordered, fixed cords, each allowed to have any number of beads. Beadless necklaces as well as beadless cords each contribute a factor 1, hence for n=0 one has 1. See A000255 for the description of a fixed cord with beads. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - _Wolfdieter Lang_, Jun 02 2010
%H Indranil Ghosh, <a href="/A086764/b086764.txt">Rows 0..50, flattened</a>
%H W. Y. C. Chen et al., <a href="http://dx.doi.org/10.1016/j.disc.2011.06.006">Higher-order log-concavity in Euler's difference table</a>, Discrete Math., 311 (2011), 2128-2134. (These are the numbers d^k_n.)
%H Fanja Rakotondrajao, <a href="http://www.emis.de/journals/INTEGERS/papers/h36/h36.Abstract.html">k-Fixed-Points-Permutations</a>, Integers: Electronic journal of combinatorial number theory 7 (2007) A36.
%F T(n, n) = 1; T(n+1, n) = n.
%F T(n+2, n) = A002061(n+1) = n^2 + n + 1; T(n+3, n) = n^3 + 3*n^2 + 5*n + 2.
%F T(n, k) = (k + 1)*T(n, k + 1) - T(n-1, k); T(n, n) = 1; T(n, k) = 0, if k > n.
%F T(n, k) = (n-1)*T(n-1, k) + (n-k-1)*T(n-2, k).
%F k!*T(n, k) = A068106(n, k). [corrected by _Georg Fischer_, Aug 13 2022]
%F Sum_{k>=0} T(n, k) = A003470(n+1).
%F T(n, k) = (1/k!) * Sum_{j>=0} (-1)^j*binomial(n-k, j)*(n-j)!. - _Philippe Deléham_, Jun 13 2005
%F From _Peter Bala_, Aug 14 2008: (Start)
%F The following remarks all relate to the array read as a square array: e.g.f for column k: exp(-y)/(1-y)^(k+1); e.g.f. for array: exp(-y)/(1-x-y) = (1 + x + x^2 + x^3 + ...) + (x + 2*x^2 + 3*x^3 + 4*x^4 + ...)*y + (1 + 3*x + 7*x^2 + 13*x^3 + ...)*y^2/2! + ... .
%F This table is closely connected to the constant e. The row, column and diagonal entries of this table occur in series formulas for e.
%F Row n for n >= 2: e = n!*(1/T(n,0) + (-1)^n*[1/(1!*T(n,0)*T(n,1)) + 1/(2!*T(n,1)*T(n,2)) + 1/(3!*T(n,2)*T(n,3)) + ...]). For example, row 3 gives e = 6*(1/2 - 1/(1!*2*11) - 1/(2!*11*32) - 1/(3!*32*71) - ...). See A095000.
%F Column 0: e = 2 + Sum_{n>=2} (-1)^n*n!/(T(n,0)*T(n+1,0)) = 2 + 2!/(1*2) - 3 !/(2*9) + 4!/(9*44) - ... .
%F Column k, k >= 1: e = (1 + 1/1! + 1/2! + ... + 1/k!) + 1/k!*Sum_{n >= 0} (-1)^n*n!/(T(n,k)*T(n+1,k)). For example, column 3 gives e = 8/3 + 1/6*(1/(1*3) - 1/(3*13) + 2/(13*71) - 6/(71*465) + ...).
%F Main diagonal: e = 1 + 2*(1/(1*1) - 1/(1*7) + 1/(7*71) - 1/(71*1001) + ...).
%F First subdiagonal: e = 8/3 + 5/(3*32) - 7/(32*465) + 9/(465*8544) - ... .
%F Second subdiagonal: e = 2*(1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...). See A143413.
%F Third subdiagonal: e = 3 - (2*3*5)/(2*53) + (3*4*7)/(53*1214) - (4*5*9)/(1214*30637) + ... .
%F For the corresponding results for the constants 1/e, sqrt(e) and 1/sqrt(e) see A143409, A143410 and A143411 respectively. For other arrays similarly related to constants see A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)). (End)
%F G.f. for column k is hypergeom([1,k+1],[],x/(x+1))/(x+1). - _Mark van Hoeij_, Nov 07 2011
%F T(n, k) = (n!/k!)*hypergeom([k-n], [-n], -1). - _Peter Luschny_, Oct 05 2017
%e Formatted as a square array:
%e 1 3 7 13 21 31 43 57 ... A002061;
%e 2 11 32 71 134 227 356 ... A094792;
%e 9 53 181 465 1001 1909 ... A094793;
%e 44 309 1214 3539 8544 ... A094794;
%e 265 2119 9403 30637 ... A023043;
%e 1854 16687 82508 ... A023044;
%e 14833 148329 ... A023045;
%e Formatted as a triangular array (mirror of A076731):
%e 1;
%e 0 1;
%e 1 1 1;
%e 2 3 2 1;
%e 9 11 7 3 1;
%e 44 53 32 13 4 1;
%e 265 309 181 71 21 5 1;
%e 1854 2119 1214 465 134 31 6 1;
%e 14833 16687 9403 3539 1001 227 43 7 1;
%e 133496 148329 82508 30637 8544 1909 356 57 8 1;
%t T[n_,k_]:=(1/k!)*Sum[(-1)^j*Binomial[n-k,j]*(n-j)!,{j,0,n}];Flatten[Table[T[n,k],{n,0,11},{k,0,n}]] (* _Indranil Ghosh_, Feb 20 2017 *)
%t T[n_, k_] := (n!/k!) HypergeometricPFQ[{k-n},{-n},-1];
%t Table[T[n,k], {n,0,9}, {k,0,n}] // Flatten (* _Peter Luschny_, Oct 05 2017 *)
%o (Magma)
%o A086764:= func< n,k | (&+[(-1)^j*Binomial(n-k,j)*Factorial(n-j): j in [0..n]])/Factorial(k) >;
%o [A086764(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 05 2023
%o (SageMath)
%o def A086764(n,k): return sum((-1)^j*binomial(n-k,j)*factorial(n-j) for j in range(n+1))//factorial(k)
%o flatten([[A086764(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 05 2023
%Y Columns: A000166, A000155, A000153, A000261, A001909, A001910, A176732, A176733, A176734, A176735, A176736.
%Y Mirror image of A076731.
%Y Cf. A002061, A003470, A008288, A023043, A023044, A023045, A068106.
%Y Cf. A076731, A094792, A094793, A094794, A095000, A108625, A143007.
%Y Cf. A143409, A143410, A143411, A143413.
%K easy,nonn,tabl
%O 0,7
%A _Philippe Deléham_, Aug 02 2003
%E More terms from _David Wasserman_, Mar 28 2005
%E Additional comments from _Zerinvary Lajos_, Mar 30 2006
%E Edited by _N. J. A. Sloane_, Sep 24 2011
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