A143411 -id:A143411 - cp's OEIS frontend

A086764 Triangle T(n, k), read by row, related to Euler's difference table A068106 (divide column k of A068106 by k!).

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

Offset: 0

Philippe Deléham, Aug 02 2003

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

			Formatted as a square array:
      1      3     7    13   21   31  43 57 ... A002061;
      2     11    32    71  134  227 356    ... A094792;
      9     53   181   465 1001 1909        ... A094793;
     44    309  1214  3539 8544             ... A094794;
    265   2119  9403 30637                  ... A023043;
   1854  16687 82508                        ... A023044;
  14833 148329                              ... 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;

Indranil Ghosh, Rows 0..50, flattened
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.)
Fanja Rakotondrajao, k-Fixed-Points-Permutations, Integers: Electronic journal of combinatorial number theory 7 (2007) A36.

Columns: A000166, A000155, A000153, A000261, A001909, A001910, A176732, A176733, A176734, A176735, A176736.
Mirror image of A076731.
Cf. A002061, A003470, A008288, A023043, A023044, A023045, A068106.
Cf. A076731, A094792, A094793, A094794, A095000, A108625, A143007.
Cf. A143409, A143410, A143411, A143413.

A086764:= func< n,k | (&+[(-1)^j*Binomial(n-k,j)*Factorial(n-j): j in [0..n]])/Factorial(k) >;
[A086764(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

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[n_, k_] := (n!/k!) HypergeometricPFQ[{k-n},{-n},-1];
Table[T[n,k], {n,0,9}, {k,0,n}] // Flatten (* Peter Luschny, Oct 05 2017 *)

def A086764(n,k): return sum((-1)^j*binomial(n-k,j)*factorial(n-j) for j in range(n+1))//factorial(k)
flatten([[A086764(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

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, k). [corrected by Georg Fischer, Aug 13 2022]
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 Deléham, Jun 13 2005
From Peter Bala, 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} (-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} (-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
T(n, k) = (n!/k!)*hypergeom([k-n], [-n], -1). - Peter Luschny, Oct 05 2017

More terms from David Wasserman, Mar 28 2005
Additional comments from Zerinvary Lajos, Mar 30 2006
Edited by N. J. A. Sloane, Sep 24 2011

A143409 Square array read by antidiagonals: form the Euler-Seidel matrix for the sequence {k!} and then divide column k by k!.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 16, 11, 4, 1, 65, 49, 19, 5, 1, 326, 261, 106, 29, 6, 1, 1957, 1631, 685, 193, 41, 7, 1, 13700, 11743, 5056, 1457, 316, 55, 8, 1, 109601, 95901, 42079, 12341, 2721, 481, 71, 9, 1, 986410, 876809, 390454, 116125, 25946, 4645, 694, 89, 10, 1

Offset: 0

Peter Bala, Aug 14 2008

The Euler-Seidel matrix for the sequence {k!} is array A076571 read as a square, whose k-th column entries have a common factor of k!. Removing these common factors gives the current table.
This table is closely connected to the constant 1/e. The row, column and diagonal entries of this table occur in series acceleration formulas for 1/e.
For a similar table based on the differences of the sequence {k!} and related to the constant e, see A086764. For other arrays similarly related to constants see A143410 (for sqrt(e)), A143411 (for 1/sqrt(e)), A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)).

			The Euler-Seidel matrix for the sequence {k!} begins
==============================================
n\k|.....0.....1.....2.....3.....4.....5.....6
==============================================
0..|.....1.....1.....2.....6....24...120...720
1..|.....2.....3.....8....30...144...840
2..|.....5....11....38...174...984
3..|....16....49...212..1158
4..|....65...261..1370
5..|...326..1631
6..|..1957
...
Dividing the k-th column by k! gives
==============================================
n\k|.....0.....1.....2.....3.....4.....5.....6
==============================================
0..|.....1.....1.....1.....1.....1.....1.....1
1..|.....2.....3.....4.....5.....6.....7
2..|.....5....11....19....29....41
3..|....16....49...106...193
4..|....65...261...685
5..|...326..1631
6..|..1957
...
Examples of series formula for 1/e:
Row 2: 1/e = 2*(1/5 - 1/(1!*5*11) + 1/(2!*11*19) - 1/(3!*19*29) + ...).
Column 4: 24/e = 9 - (0!/(1*6) + 1!/(6*41) + 2!/(41*316) + ...).
...
Displayed as a triangle:
0 |     1
1 |     2,     1
2 |     5,     3,    1
3 |    16,    11,    4,    1
4 |    65,    49,   19,    5,   1
5 |   326,   261,  106,   29,   6,  1
6 |  1957,  1631,  685,  193,  41,  7, 1
7 | 13700, 11743, 5056, 1457, 316, 55, 8, 1

Robert Israel, Table of n, a(n) for n = 0..10010 (antidiagonals 0 to 140, flattened)
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Eric Weisstein's World of Mathematics Poisson-Charlier polynomial

Cf. A008288, A076571, A086764, A108625, A143007, A143410, A143411, A143413, A001517 (main diagonal), A028387 (row 2), A000522 (column 0), A001339 (column 1), A082030 (column 2), A095000 (column 3), A095177 (column 4).

Cf. A273596, A003470.

T := (n, k) -> 1/k!*add(binomial(n,j)*(k+j)!, j = 0..n):
for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;
# Alternate:
T:= proc(n,k) option remember;
  if n = 0 then return 1 fi;
  (n+k)*procname(n-1,k) + procname(n-1,k-1);
end proc:
seq(seq(T(s-n,n),n=0..s),s=0..10); # Robert Israel, Jul 07 2017
# Or:
A143409 := (n,k) -> hypergeom([k+1, k-n], [], -1):
seq(seq(simplify(A143409(n,k)),k=0..n),n=0..9); # Peter Luschny, Oct 05 2017

T[n_, k_] := HypergeometricPFQ[{k+1,k-n}, {}, -1];
Table[T[n,k], {n,0,9}, {k,0,n}] // Flatten (* Peter Luschny, Oct 05 2017 *)

T(n,k) = (1/k!)*Sum_{j = 0..n} binomial(n,j)*(k+j)!.
T(n,k) = ((n+k)!/k!)*Num_Pade(n,k), where Num_Pade(n,k) denotes the numerator of the Padé approximation for the function exp(x) of degree (n,k) evaluated at x = 1.
Recurrence relations:
T(n,k) = T(n-1,k) + (k+1)*T(n-1,k+1);
T(n,k) = (n+k)*T(n-1,k) + T(n-1,k-1).
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 + ...) + (2 + 3*x + 4*x^2 + ...)*y + (5 + 11*x + 19*x^2 + ...)*y^2/2! + ... .
Row n lists the values of the Poisson-Charlier polynomial x^(n) + C(n,1)*x^(n-1) + C(n,2)*x^(n-2) + ... + C(n,n) for x = 1,2,3,..., where x^(m) denotes the rising factorial x*(x+1)*...*(x+m-1).
Main diagonal is A001517.
Series formulas for 1/e:
Row n: 1/e = n!*[1/T(n,0) - 1/(1!*T(n,0)*T(n,1)) + 1/(2!*T(n,1)*T(n,2)) - 1/(3!*T(n,2)*T(n,3)) + ...].
Column k: k!/e = A000166(k) + (-1)^(k+1)*[0!/(T(0,k)*T(1,k)) + 1!/(T(1,k)*T(2,k)) + 2!/(T(2,k)*T(3,k)) + ...].
Main diagonal: 1/e = 1 - 2*Sum_{n>=0} (-1)^n/(T(n,n)*T(n+1,n+1)) = 1 - 2*[1/(1*3) - 1/(3*19) + 1/(19*193) - ...].
Second subdiagonal: 1/e = 2*(1^2/(1*5) - 2^2/(5*49) + 3^2/(49*685) - ...).
Compare with A143413.
From Peter Luschny, Oct 05 2017: (Start)
T(n, k) = hypergeom([k+1, k-n], [], -1).
When seen as a triangular array then the row sums are A273596 and the alternating row sums are A003470. (End)

A065919 Bessel polynomial y_n(4).

Original entry on oeis.org

1, 5, 61, 1225, 34361, 1238221, 54516085, 2836074641, 170218994545, 11577727703701, 880077524475821, 73938089783672665, 6803184337622361001, 680392371852019772765, 73489179344355757819621, 8525425196317119926848801, 1057226213522667226687070945

Offset: 0

N. J. A. Sloane, Dec 08 2001

Main diagonal of A143411. - Peter Bala, Aug 14 2008

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Harry J. Smith, Table of n, a(n) for n = 0..100
W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001517, A001518.
Cf. A143411 (main diagonal), A143412.
Polynomial coefficients are in A001498.

A065919:= func< n | (&+[Binomial(n,k)*Factorial(n+k)*2^k/Factorial(n): k in [0..n]]) >;
[A065919(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

seq(simplify(2^n*KummerU(-n,-2*n,1/2)), n=0..16); # Peter Luschny, May 10 2022

Table[Sum[(n+k)!*2^k/((n-k)!*k!), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)

for (n=0, 100, if (n>1, a=4*(2*n - 1)*a1 + a2; a2=a1; a1=a, if (n, a=a1=5, a=a2=1)); write("b065919.txt", n, " ", a) ) \\ Harry J. Smith, Nov 04 2009

a(n) = sum(k=0,n, (n+k)!*2^k/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013

def A065919(n): return sum(binomial(n,k)*factorial(n+k)*2^k/factorial(n) for k in range(n+1))
[A065919(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
From Peter Bala, Aug 14 2008: (Start)
Recurrence relation: a(0) = 1, a(1) = 5, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A143412(n) satisfies the same recurrence relation.
1/sqrt(e) = 1 - 2*Sum_{n = 0..inf} (-1)^n/(a(n)*a(n+1)) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). (End)
G.f.: 1/Q(0), where Q(k)= 1 - x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = exp(1/4)/sqrt(2*Pi)*BesselK(n+1/2,1/4). - Gerry Martens, Jul 22 2015
a(n) ~ 2^(3*n+1/2) * n^n / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015
From Peter Bala, Apr 12 2017: (Start)
a(n) = 1/n!*Integral_{x = 0..inf} x^n*(1 + 2*x)^n dx.
E.g.f.: d/dx( exp(x*c(2*x)) ) = 1 + 5*x + 61*x^2/2! + 1225*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
G.f.: (1/(1-x))*hypergeometric2f0(1,1/2; - ; 8*x/(1-x)^2). - G. C. Greubel, Aug 16 2017
a(n) = 2^n*KummerU(-n, -2*n, 1/2). - Peter Luschny, May 10 2022

A143410 Form the difference table of the sequence {2^kk!}, then divide k-th column entries by 2^kk!. Array read by ascending antidiagonals, T(n, k) for n, k >= 0.

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 29, 17, 5, 1, 233, 131, 37, 7, 1, 2329, 1281, 353, 65, 9, 1, 27949, 15139, 4105, 743, 101, 11, 1, 391285, 209617, 56189, 10049, 1349, 145, 13, 1, 6260561, 3325923, 883885, 156679, 20841, 2219, 197, 15, 1, 112690097, 59475329, 15700313

Offset: 0

Peter Bala, Aug 19 2008

This table is closely connected to the constant sqrt(e). The row, column and diagonal entries of this table occur in series acceleration formulas for sqrt(e). For a similar table based on the Euler-Seidel matrix of the sequence {2^k*k!} and related to the constant 1/sqrt(e), see A143411. For other arrays similarly related to constants see A086764 (for e), A143409 (for 1/e), A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)).

			Table of differences of {2^k*k!}
  =====================================================
  Column                0     1     2     3     4     5
  =====================================================
  Sequence 2^k*k!       1     2     8    48   384  3840
  First differences     1     6    40   336  3456
  Second differences    5    34   296  3120
  Third differences    29   262  2824
  Fourth differences  233  2562
  ...
Remove the common factor 2^k*k! from k-th column entries:
  ====================================
  n\k|   0      1      2      3      4
  ====================================
  0  |   1      1      1      1      1
  1  |   1      3      5      7      9
  2  |   5     17     37     65    101
  3  |  29    131    353    743   1349
  4  | 233   1281   4105  10049  20841

Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial.

Cf. A008288, A076571, A086764, A108625, A143007, A143409, A143411, A143412.

T := (n, k) -> (-1)^n/k!*add((-2)^j*binomial(n,j)*(k+j)!, j = 0..n):
for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;

T(n,k) = ((-1)^n/k!)*Sum {j = 0..n} (-2)^j*C(n,j)*(k+j)!.
Relation with Poisson-Charlier polynomials c_n(x,a): T(n,k) = c_n(-(k+1),-1/2).
Recurrence relations: T(n,k) = 2*n*T(n-1,k) + T(n,k-1); T(n,k) = 2*(n+k)*T(n-1,k) - T(n-1,k-1); T(n,k) = 2*(k+1)*T(n-1,k+1) - T(n-1,k);
Recurrence for row n entries: 2*k*T(n,k) = (2*n+2*k+1)*T(n,k-1) - T(n,k-2).
E.g.f. for column k: exp(-y)/(1-2*y)^(k+1).
E.g.f. for array: exp(-y)/(1-x-2*y) = (1 + x + x^2 + ...) + (1 + 3*x + 5*x^2 + ...)*y + (5 + 17*x + 37*x^2 + ...)*y^2/2! + ... .
Series acceleration formulas for sqrt(e):
Row n: sqrt(e) = 2^n*n!*(1/T(n,0) + (-1)^n*(1/(2*1!*T(n,0)*T(n,1)) + 1/(2^2*2!*T(n,1)*T(n,2)) + 1/(2^3*3!*T(n,2)*T(n,3)) + ...)). For example, row 3 gives sqrt(e) = 48*(1/29 - 1/(2*29*131) - 1/(8*131*353) - 1/(48*353*743) - ...).
Column k: sqrt(e) = (1 + (1/2)/1! + (1/2)^2 / 2! + ... + (1/2)^k/k!) + 1/(2^k*k!) * Sum_{n>= 0} ((-2)^n *n!/(T(n,k)*T(n+1,k))). For example, column 3 gives sqrt(e) = 79/48 + (1/48)*(1/(1*7) - 2/(7*65) + 8/(65*743) - 48/(743*10049) + ...).
Main diagonal: sqrt(e) = 1 + 2*(1/(1*3) - 1/(3*37) + 1/(37*743) - ...). See A143412.
T(n, k) = (-1)^n*(-1/2)^(k + 1)*KummerU(k + 1, k + n + 2, -1/2). - Peter Luschny, Jan 02 2020

cp's OEIS Frontend

A086764 Triangle T(n, k), read by row, related to Euler's difference table A068106 (divide column k of A068106 by k!).

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A143409 Square array read by antidiagonals: form the Euler-Seidel matrix for the sequence {k!} and then divide column k by k!.

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A065919 Bessel polynomial y_n(4).

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A143410 Form the difference table of the sequence {2^k*k!}, then divide k-th column entries by 2^k*k!. Array read by ascending antidiagonals, T(n, k) for n, k >= 0.

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A143410 Form the difference table of the sequence {2^kk!}, then divide k-th column entries by 2^kk!. Array read by ascending antidiagonals, T(n, k) for n, k >= 0.