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A154921 Triangle read by rows, T(n, k) = binomial(n, k) * Sum_{j=0..n-k} E(n-k, j)*2^j, where E(n, k) are the Eulerian numbers A173018(n, k), for 0 <= k <= n.

%I A154921 #71 Jun 23 2023 05:38:13
%S A154921 1,1,1,3,2,1,13,9,3,1,75,52,18,4,1,541,375,130,30,5,1,4683,3246,1125,
%T A154921 260,45,6,1,47293,32781,11361,2625,455,63,7,1,545835,378344,131124,
%U A154921 30296,5250,728,84,8,1,7087261,4912515,1702548,393372,68166,9450,1092,108,9,1
%N A154921 Triangle read by rows, T(n, k) = binomial(n, k) * Sum_{j=0..n-k} E(n-k, j)*2^j, where E(n, k) are the Eulerian numbers A173018(n, k), for 0 <= k <= n.
%C A154921 Previous name: Matrix inverse of A154926.
%C A154921 A000670 appears in the first column. A052882 appears in the second column. A000027 and A045943 appear as diagonals. An alternative to calculating the matrix inverse of A154926 is to move the term in the lower right corner to a position in the same column and calculate the determinant instead, which yields the same answer.
%C A154921 Matrix inverse of (2*I - P), where P is Pascal's triangle and I the identity matrix. See A162312 for the matrix inverse of (2*P - I) and some general remarks about arrays of the form M(a) := (I - a*P)^-1 and their connection with weighted sums of powers of integers. The present array equals (1/2)*M(1/2). - _Peter Bala_, Jul 01 2009
%C A154921 From _Mats Granvik_, Aug 11 2009: (Start)
%C A154921 The values in this triangle can be seen as permanents of the Pascal triangle analogous to the method in the Redheffer matrix. The elements satisfy (T(n,k)/T(n,k-1))*k = (T(n-1,k)/T(n,k))*n which converges to log(2) as n->oo and k->0. More generally to calculate log(x) multiply the negative values in A154926 by 1/(x-1) and calculate the matrix inverse. Then (T(n,k)/T(n,k-1))*k and (T(n-1,k)/T(n,k))*n in the resulting triangle converge to log(x).
%C A154921 This method for calculating log(x) converges faster than the Taylor series when x is greater than 5 or so. See chapter on Taylor series in Spiegel for comparison. (End)
%C A154921 Exponential Riordan array [1/(2-exp(x)),x]. - _Paul Barry_, Apr 06 2011
%C A154921 T(n,k) is the number of ordered set partitions of {1,2,...,n} such that the first block contains k elements.  For k=0 the first block contains arbitrarily many elements. - _Geoffrey Critzer_, Jul 22 2013
%C A154921 A natural (signed) refinement of these polynomials is given by the Appell sequence e.g.f. e^(xt)/ f(t) = exp[tP.(x)] with the formal Taylor series f(x) = 1 + x[1] x + x[2] x^2/2! + ... and with raising operator R = x - d[log(f(D)]/dD (cf. A263634). - _Tom Copeland_, Nov 06 2015
%D A154921 Murray R. Spiegel, Mathematical handbook, Schaum's Outlines, p. 111.
%H A154921 Alois P. Heinz, <a href="/A154921/b154921.txt">Rows n = 0..140, flattened</a>
%H A154921 R. B. Nelsen, <a href="http://www.jstor.org/stable/2323104">Problem E3062</a>, Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377.
%H A154921 R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in Power Sets</a>, Mathematics Magazine, Vol. 64, No. 1 (Feb., 1991), 23-31.
%F A154921 From _Peter Bala_, Jul 01 2009: (Start)
%F A154921 TABLE ENTRIES
%F A154921   (1) T(n,k) = binomial(n,k)*A000670(n-k).
%F A154921 GENERATING FUNCTION
%F A154921   (2) exp(x*t)/(2-exp(t)) = 1 + (1+x)*t + (3+2*x+x^2)*t^2/2! + ....
%F A154921 PROPERTIES OF THE ROW POLYNOMIALS
%F A154921 The row generating polynomials R_n(x) form an Appell sequence. They appear in the study of the poset of power sets [Nelsen and Schmidt].
%F A154921 The first few values are R_0(x) = 1, R_1(x) = 1+x, R_2(x) = 3+2*x+x^2 and R_3(x) = 13+9*x+3*x^2+x^3.
%F A154921 The row polynomials may be recursively computed by means of
%F A154921   (3) R_n(x) = x^n + Sum_{k = 0..n-1} binomial(n,k)*R_k(x).
%F A154921 Explicit formulas include
%F A154921   (4) R_n(x) = (1/2)*Sum_{k >= 0} (1/2)^k*(x+k)^n,
%F A154921   (5) R_n(x) = Sum_{j = 0..n} Sum_{k = 0..j} (-1)^(j-k)*binomial(j,k) *(x+k)^n,
%F A154921 and
%F A154921   (6) R_n(x) = Sum_{j = 0..n} Sum_{k = j..n} k!*Stirling2(n,k) *binomial(x,k-j).
%F A154921 SUMS OF POWERS OF INTEGERS
%F A154921 The row polynomials satisfy the difference equation
%F A154921   (7) 2*R_m(x) - R_m(x+1) = x^m,
%F A154921 which easily leads to the evaluation of the weighted sums of powers of integers
%F A154921   (8) Sum_{k = 1..n-1} (1/2)^k*k^m = 2*R_m(0) - (1/2)^(n-1)*R_m(n).
%F A154921 For example, m = 2 gives
%F A154921   (9) Sum_{k = 1..n-1} (1/2)^k*k^2 = 6 - (1/2)^(n-1)*(n^2+2*n+3).
%F A154921 More generally we have
%F A154921   (10) Sum_{k=0..n-1} (1/2)^k*(x+k)^m = 2*R_m(x) - (1/2)^(n-1)*R_m(x+n).
%F A154921 RELATIONS WITH OTHER SEQUENCES
%F A154921 Sequences in the database given by particular values of the row polynomials are
%F A154921   (11) A000670(n) = R_n(0)
%F A154921   (12) A052841(n) = R_n(-1)
%F A154921   (13) A000629(n) = R_n(1)
%F A154921   (14) A007047(n) = R_n(2)
%F A154921   (15) A080253(n) = 2^n*R_n(1/2).
%F A154921 This last result is the particular case (x = 0) of the result that the polynomials 2^n*R_n(1/2+x/2) are the row generating polynomials for A162313.
%F A154921 The above formulas should be compared with those for A162312. (End)
%F A154921 From _Peter Luschny_, Jul 15 2012: (Start)
%F A154921   (16) A151919(n) = R_n(1/3)*3^n*(-1)^n
%F A154921   (17) A052882(n) = [x^1] R_n(x)
%F A154921   (18) A045943(n) = [x^(n-1)] R_n+1(x)
%F A154921   (19) A099880(n) = [x^n] R_2n(x). (End)
%F A154921 The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1} binomial(n,k)*p{k}(0)*(1+x^(n-k)). - _Peter Luschny_, Jul 15 2012
%e A154921 From _Peter Bala_, Jul 01 2009: (Start)
%e A154921 Triangle T(n, k) begins:
%e A154921 n\k|     0     1     2     3     4     5     6
%e A154921 ==============================================
%e A154921 0  |     1
%e A154921 1  |     1     1
%e A154921 2  |     3     2     1
%e A154921 3  |    13     9     3     1
%e A154921 4  |    75    52    18     4     1
%e A154921 5  |   541   375   130    30     5     1
%e A154921 6  |  4683  3246  1125   260    45     6     1
%e A154921 ...
%e A154921 (End)
%e A154921 From _Mats Granvik_, Aug 11 2009: (Start)
%e A154921 Row 4 equals 75,52,18,4,1 because permanents of:
%e A154921   1,0,0,0,1  1,0,0,0,0  1,0,0,0,0  1,0,0,0,0  1,0,0,0,0
%e A154921   1,1,0,0,0  1,1,0,0,1  1,1,0,0,0  1,1,0,0,0  1,1,0,0,0
%e A154921   1,2,1,0,0  1,2,1,0,0  1,2,1,0,1  1,2,1,0,0  1,2,1,0,0
%e A154921   1,3,3,1,0  1,3,3,1,0  1,3,3,1,0  1,3,3,1,1  1,3,3,1,0
%e A154921   1,4,6,4,0  1,4,6,4,0  1,4,6,4,0  1,4,6,4,0  1,4,6,4,1
%e A154921 are:
%e A154921      75         52         18          4          1
%e A154921 (End)
%p A154921 A154921_row := proc(n) local i,p; p := proc(n,x) option remember; local k;
%p A154921 if n = 0 then 1 else add(p(k,0)*binomial(n,k)*(1+x^(n-k)),k=0..n-1) fi end:
%p A154921 seq(coeff(p(n,x),x,i),i=0..n) end: for n from 0 to 5 do A154921_row(n) od;
%p A154921 # _Peter Luschny_, Jul 15 2012
%p A154921 T := (n,k) -> binomial(n,k)*add(combinat:-eulerian1(n-k,j)*2^j, j=0..n-k):
%p A154921 seq(print(seq(T(n,k), k=0..n)),n=0..6); # _Peter Luschny_, Feb 07 2015
%p A154921 # third Maple program:
%p A154921 b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
%p A154921 T:= (n, k)-> n!/k! *b(n-k):
%p A154921 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Feb 03 2019
%p A154921 # fourth Maple program:
%p A154921 p := proc(n, m) option remember; if n = 0 then 1 else
%p A154921     (m + x)*p(n - 1, m) + (m + 1)*p(n - 1, m + 1) fi end:
%p A154921 row := n -> local k; seq(coeff(p(n, 0), x, k), k = 0..n):
%p A154921 for n from 0 to 6 do row(n) od;  # _Peter Luschny_, Jun 23 2023
%t A154921 nn = 8; a = Exp[x] - 1;
%t A154921 Map[Select[#, # > 0 &] &,
%t A154921   Transpose[
%t A154921    Table[Range[0, nn]! CoefficientList[
%t A154921 Series[x^n/n!/(1 - a), {x, 0, nn}], x], {n, 0, nn}]]] // Grid (* _Geoffrey Critzer_, Jul 22 2013 *)
%t A154921 E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k < 0 || k > n = 0; E1[n_, k_] := E1[n, k] = (n - k) E1[n - 1, k - 1] + (k + 1) E1[n - 1, k];
%t A154921 T[n_, k_] := Binomial[n, k] Sum[E1[n - k, j] 2^j, {j, 0, n - k}];
%t A154921 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 30 2018, after _Peter Luschny_ *)
%o A154921 (Sage)
%o A154921 @CachedFunction
%o A154921 def Poly(n, x):
%o A154921     return 1 if n == 0 else add(Poly(k,0)*binomial(n,k)*(x^(n-k)+1) for k in range(n))
%o A154921 R = PolynomialRing(ZZ, 'x')
%o A154921 for n in (0..6): print(R(Poly(n,x)).list()) # _Peter Luschny_, Jul 15 2012
%Y A154921 Cf. A000629 (row sums), A000670, A007047, A052822 (column 1), A052841 (alt. row sums), A080253, A162312, A162313.
%Y A154921 Cf. A263634, A099880 (T(2n,n)).
%K A154921 nonn,tabl
%O A154921 0,4
%A A154921 _Mats Granvik_, Jan 17 2009
%E A154921 New name by _Peter Luschny_, Feb 07 2015