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A086764
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Triangle T(n, k), read by row, related to Euler's difference table A068106 (divide k-th diagonal of A068106 by k!).
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19
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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, 1854, 2119, 1214, 465, 134, 31, 6, 1, 14833, 16687, 9403, 3539, 1001, 227, 43, 7, 1, 133496, 148329, 82508, 30637, 8544, 1909, 356, 57, 8, 1, 1334961, 1468457, 808393
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| 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 a beadless cords contribute each 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 ( Febr 27 2010). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 02 2010]
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REFERENCES
| W. Y. C. Chen et al., Higher-order log-concavity in Euler's difference table, Discrete Math., 311 (2011), 2128-2134. (These are the numbers d^k_n.)
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FORMULA
| T(n, n)= 1; T(n+1, n)= n; T(n+2, n)= A002061(n+1)= n^2 + n + 1; T(n+3, n)= n^3 + 3*n^2 + 5*n + 2.
T(n, k) = (k + 1)*T(n, k + 1)-T(n-1, k); T(n, n)= 1; T(n, k)= 0, if k>n. T(n, k) = (n-1)*T(n-1, k) + (n-k-1)*T(n-2, k). k!*T(n, k) = A068106(n+1, k+1). Sum(k>=0; T(n, k) = A003470(n+1).
T(n, k) = 1/k! * Sum_{j>=0} (-1)^j*binomial(n-k, j)*(n-j)!. - Philippe DELEHAM, Jun 13 2005
Comments from Peter Bala (pbala(AT)toucansurf.com), Aug 14 2008 (Start): 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! + ... .
This table is closely connected to the constant e. The row, column and diagonal entries of this table occur in series formulas for e.
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.
Column 0: e = 2 + sum {n = 2..inf} (-1)^n*n!/(T(n,0)*T(n+1,0)) = 2 + 2!/(1*2) - 3 !/(2*9) + 4!/(9*44) - ... .
Column k, k >= 1: e = (1+1/1!+1/2!+...+1/k!)+ 1/k!*sum {n = 0..inf} (-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) + ...].
Main diagonal: e = 1 + 2*[1/(1*1) - 1/(1*7) + 1/(7*71) - 1/(71*1001) + ...].
First subdiagonal: e = 8/3 + 5/(3*32) - 7/(32*465) + 9/(465*8544) - ... .
Second subdiagonal: e = 2*[1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...]. See A143413.
Third subdiagonal: e = 3 - (2*3*5)/(2*53) + (3*4*7)/(53*1214) - (4*5*9)/(1214*30637) + ... .
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)
G.f. for column k is hypergeom([1,k+1],[],x/(x+1))/(x+1) - Mark van Hoeij, Nov 07 2011
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EXAMPLE
| 1; 0, 1; 1, 1, 1; 2, 3, 2, 1; 9, 11, 7, 3, 1; 44, 53, 32, 13, 4, 1; ...
Formatted as a square array:
1 3 7 13 21 31 43 57 which equals A002061
2 11 32 71 134 227 356 which equals A094792
9 53 181 465 1001 1909 which equals A094793
44 309 1214 3539 8544 which equals A094794
265 2119 9403 30637 which equals A023043
1854 16687 82508 which equals A023044
14833 148329 which equals A023045
Formatted as a triangular array (mirror of A076731):
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
1854 2119 1214 465 134 31 6 1
14833 16687 9403 3539 1001 227 43 7 1
133496 148329 82508 30637 8544 1909 356 57 8 1
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CROSSREFS
| Columns: A000166, A000155, A000153, A000261, A001909, A001910, A176732 - A176736.
Cf. A068106, A003470, A002061. Mirror image of A076731.
Cf. A143409, A143410, A143411, A143413.
Sequence in context: A062323 A020858 A090664 * A076224 A114729 A186037
Adjacent sequences: A086761 A086762 A086763 * A086765 A086766 A086767
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 02 2003
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EXTENSIONS
| More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Mar 28 2005
Additional comments from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 30 2006
Edited by N. J. A. Sloane, Sep 24 2011
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