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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!.

%I A143409 #25 Feb 16 2025 08:33:08
%S A143409 1,2,1,5,3,1,16,11,4,1,65,49,19,5,1,326,261,106,29,6,1,1957,1631,685,
%T A143409 193,41,7,1,13700,11743,5056,1457,316,55,8,1,109601,95901,42079,12341,
%U A143409 2721,481,71,9,1,986410,876809,390454,116125,25946,4645,694,89,10,1
%N A143409 Square array read by antidiagonals: form the Euler-Seidel matrix for the sequence {k!} and then divide column k by k!.
%C A143409 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.
%C A143409 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.
%C A143409 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)).
%H A143409 Robert Israel, <a href="/A143409/b143409.txt">Table of n, a(n) for n = 0..10010</a> (antidiagonals 0 to 140, flattened)
%H A143409 D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%H A143409 Eric Weisstein's World of Mathematics <a href="https://mathworld.wolfram.com/Poisson-CharlierPolynomial.html">Poisson-Charlier polynomial</a>
%F A143409 T(n,k) = (1/k!)*Sum_{j = 0..n} binomial(n,j)*(k+j)!.
%F A143409 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.
%F A143409 Recurrence relations:
%F A143409 T(n,k) = T(n-1,k) + (k+1)*T(n-1,k+1);
%F A143409 T(n,k) = (n+k)*T(n-1,k) + T(n-1,k-1).
%F A143409 E.g.f. for column k: exp(y)/(1-y)^(k+1).
%F A143409 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! + ... .
%F A143409 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).
%F A143409 Main diagonal is A001517.
%F A143409 Series formulas for 1/e:
%F A143409 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)) + ...].
%F A143409 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)) + ...].
%F A143409 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) - ...].
%F A143409 Second subdiagonal: 1/e = 2*(1^2/(1*5) - 2^2/(5*49) + 3^2/(49*685) - ...).
%F A143409 Compare with A143413.
%F A143409 From _Peter Luschny_, Oct 05 2017: (Start)
%F A143409 T(n, k) = hypergeom([k+1, k-n], [], -1).
%F A143409 When seen as a triangular array then the row sums are A273596 and the alternating row sums are A003470. (End)
%e A143409 The Euler-Seidel matrix for the sequence {k!} begins
%e A143409 ==============================================
%e A143409 n\k|.....0.....1.....2.....3.....4.....5.....6
%e A143409 ==============================================
%e A143409 0..|.....1.....1.....2.....6....24...120...720
%e A143409 1..|.....2.....3.....8....30...144...840
%e A143409 2..|.....5....11....38...174...984
%e A143409 3..|....16....49...212..1158
%e A143409 4..|....65...261..1370
%e A143409 5..|...326..1631
%e A143409 6..|..1957
%e A143409 ...
%e A143409 Dividing the k-th column by k! gives
%e A143409 ==============================================
%e A143409 n\k|.....0.....1.....2.....3.....4.....5.....6
%e A143409 ==============================================
%e A143409 0..|.....1.....1.....1.....1.....1.....1.....1
%e A143409 1..|.....2.....3.....4.....5.....6.....7
%e A143409 2..|.....5....11....19....29....41
%e A143409 3..|....16....49...106...193
%e A143409 4..|....65...261...685
%e A143409 5..|...326..1631
%e A143409 6..|..1957
%e A143409 ...
%e A143409 Examples of series formula for 1/e:
%e A143409 Row 2: 1/e = 2*(1/5 - 1/(1!*5*11) + 1/(2!*11*19) - 1/(3!*19*29) + ...).
%e A143409 Column 4: 24/e = 9 - (0!/(1*6) + 1!/(6*41) + 2!/(41*316) + ...).
%e A143409 ...
%e A143409 Displayed as a triangle:
%e A143409 0 |     1
%e A143409 1 |     2,     1
%e A143409 2 |     5,     3,    1
%e A143409 3 |    16,    11,    4,    1
%e A143409 4 |    65,    49,   19,    5,   1
%e A143409 5 |   326,   261,  106,   29,   6,  1
%e A143409 6 |  1957,  1631,  685,  193,  41,  7, 1
%e A143409 7 | 13700, 11743, 5056, 1457, 316, 55, 8, 1
%p A143409 T := (n, k) -> 1/k!*add(binomial(n,j)*(k+j)!, j = 0..n):
%p A143409 for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;
%p A143409 # Alternate:
%p A143409 T:= proc(n,k) option remember;
%p A143409   if n = 0 then return 1 fi;
%p A143409   (n+k)*procname(n-1,k) + procname(n-1,k-1);
%p A143409 end proc:
%p A143409 seq(seq(T(s-n,n),n=0..s),s=0..10); # _Robert Israel_, Jul 07 2017
%p A143409 # Or:
%p A143409 A143409 := (n,k) -> hypergeom([k+1, k-n], [], -1):
%p A143409 seq(seq(simplify(A143409(n,k)),k=0..n),n=0..9); # _Peter Luschny_, Oct 05 2017
%t A143409 T[n_, k_] := HypergeometricPFQ[{k+1,k-n}, {}, -1];
%t A143409 Table[T[n,k], {n,0,9}, {k,0,n}] // Flatten (* _Peter Luschny_, Oct 05 2017 *)
%Y A143409 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).
%Y A143409 Cf. A273596, A003470.
%K A143409 easy,nonn,tabl
%O A143409 0,2
%A A143409 _Peter Bala_, Aug 14 2008