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 A239473 Triangle read by rows: signed version of A059260: coefficients for expansion of partial sums of sequences a(n,x) in terms of their binomial transforms (1+a(.,x))^n ; Laguerre polynomial expansion of the truncated exponential. 7
 1, 0, 1, 1, -1, 1, 0, 2, -2, 1, 1, -2, 4, -3, 1, 0, 3, -6, 7, -4, 1, 1, -3, 9, -13, 11, -5, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS With T the lower triangular array above and the Laguerre polynomials L(k,x)=sum{j=0..k, (-1)^j binomial(k, j) x^j/j!}, the following identities hold: A) sum{k=0..n, (-1)^k L(k,x)}= sum{k=0..n, T(n,k) x^k/k!} B) sum{k=0..n, x^k/k!}= sum{k=0..n, T(n,k) L(k,-x)} C) sum{k=0..n, x^k}= sum{k=0..n, T(n,k) (1+x)^k} = (1-x^(n+1))/(1-x). More generally, for polynomial sequences, D) sum{k=0..n, P(k,x)}= sum{k=0..n, T(n,k) (1+P(.,x))^k}, where, e.g., for an Appell sequence, such as the  Bernoulli polynomials, umbrally, (1+ Ber(.,x))^k = Ber(k,x+1). Identity B follows from A through umbral substitution of j!L(j,-x) for x^j in A. Identity C, related to the cyclotomic polynomials for prime index, follows from B through the Laplace transform. Integrating C gives sum{k=0..n,  T(n,k) (2^(k+1)-1)/(k+1)}= H(n+1), the harmonic numbers. Identity A>=0 for x>=0 (see MathOverflow link for evaluation in terms of Hermite polynomials). From identity C, W(m,n)= (-1)^n sum{k=0..n, T(n,k) (2-m)^k}= number of walks of length n+1 between any two distinct vertices of the complete graph K_m for m>2. LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened J. Adams, On the groups J(x)-II, Topology, Vol. 3, p. 137-171, Pergamon Press, (1965). MathOverflow, Cyclotomic Polynomials in Combinatorics Mathoverflow, Inequality for Laguerre-polynomials FORMULA T(n, k) = Sum{j=0..n} (-1)^(j+k) * binomial(j, k). E.g.f: (exp(t)-(x-1)*exp((x-1)*t))/(2-x). O.g.f. (n-th row): (1-(x-1)^(n+1))/(2-x). Associated operator identities: With D=d/dx, :xD:^n=x^n*D^n, and :Dx:^n=D^n*x^n, then bin(xD,n)= binomial(xD,n)=:xD:^n/n! and L(n,-:xD:)=:Dx:^n/n!=bin(xD+n,n)=(-1)^n bin(-xD-1,n), A-o) sum{k=0..n, (-1)^k L(k,-:xD:)}= sum{k=0..n, :-Dx:^k/k!}      =sum{k=0..n, T(n,k) :-xD:^k/k!}= sum{k=0..n, (-1)^k T(n,k)bin(xD,k)} B-o) sum{k=0..n, :xD:^k/k!}= sum{k=0..n, T(n,k) L(k,-:xD:)}      = sum{k=0..n, T(n,k) :Dx:^k/k!}= sum{k=0..n, bin(xD,k)}. Associated binomial identities: A-b) sum{k=0..n, (-1)^k bin(s+k,k)}= sum{k=0..n, (-1)^k T(n,k) bin(s,k)}      = sum{k=0..n, bin(-s-1,k)}= sum{k=0..n, T(n,k) bin(-s-1+k,k)} B-b) sum{k=0..n, bin(s,k)}= sum{k=0..n, T(n,k) bin(s+k,k)}      = sum{k=0..n, (-1)^k bin(-s-1+k,k)}      = sum{k=0..n, (-1)^k T(n,k) bin(-s-1,k)}. In particular, from B-b with s=n, sum{k=0..n, T(n,k) bin(n+k,k)}= 2^n.  From B-b with s=0, row sums are all 1. From identity C with x=-2, the unsigned row sums are the Jacobsthal sequence, i.e., sum{k=0..n, T(n,k) (1+(-2))^k}= (-1)^n A001045(n+1); for x=2, the Mersenne numbers A000225; for x=-3, A014983 or signed A015518; for x=3, A003462; for x=-4, A014985 or signed A015521; for x=4, A002450; for x=-5, A014986 or signed A015531; and for x=5, A003463; for x=-6, A014987 or signed A015540; and for x=6, A003464. With -s-1=m=0,1,2,... , B-b gives finite differences (recursions): sum{k=0..n, (-1)^k T(n,k)  bin(m,k)}= sum{k=0..n,  (-1)^k  bin(m+k,k)}= T(n+m,m), i.e., finite differences of the columns of T generate shifted columns of T. The columns of T are signed, shifted versions of sequences listed in the cross-references. Since the finite difference is an involution, T(n,k)= sum{j=0..k, (-1)^j T(n+j,j)  bin(k,j)}. Gauss-Newton interpolation can be applied to give a generalized T(n,s) for s noninteger. From identity C, S(n,m)=sum{k=0..n, T(n,k) bin(k,m)}= 1 for m=0, [Ber(k+1,n+1) - Ber(k+1,0)]/(k+1) * x^k/k!), where Ber(n,x) are the Bernoulli polynomials (cf. Adams p. 140). Evaluating (d/dx)^m at x=0 of these expressions gives relations among the partial sums of the m-th powers of the integers, their binomial transforms, and the Bernoulli polynomials. With a(n,x) = (-1)^n e^(nx), the partial sums are 1-e^x+...+(-1)^n e^(nx) = sum{k=0..n, T(n,k) (1-e^x)^k } = [x/(e^x+1)] [e^((n+1)x) -1 ]/x] = [ [2x/(e^x+1)] e^((n+1)x) -  [2x/(e^x+1)] /2x = (1/2) sum(k >=0, [Eul(k+1,n+1) - Eul(k+1,0)]/(k+1) * x^k/k!), where Eul(n,x) are the Euler polynomials. Evaluating (d/dx)^m at x=0 of these expressions gives relations among the partial sums of signed m-th powers of the integers; their binomial transforms, related to the Stirling numbers of the second kind and face numbers of the permutahedra; and the Euler polynomials. (End) As in A059260, a generator in terms of bivariate polynomials with the coefficients of this entry is given by  (1/(1-y)) 1 / [1 + (y/(1-y)) x - (1/(1-y)) x^2] = 1 + y + (x^2 - x*y + y^2) + (2 x^2*y - 2 x*y^2 + y^3) + (x^4 - 2 x^3*y + 4 x^2*y^2 - 3 x*y^3 + y^4) + ... . This is of the form -h2 * 1 / (1 + h1 x + h2 x^2), related to the bivariate generator of A049310 with h1 = y/(1-y) and h2 = -1/(1-y) = -(1+h1). - Tom Copeland, Feb 16 2016 From Tom Copeland, Sep 05 2016: Letting P(k,x) = x in D gives sum_{k = 0,..,n} T(n,k) sum_{j = 0,..,k} binomial(k,j) = sum_{k = 0,..,n} T(n,k) 2^k = n + 1. The quantum integers [n+1]_q = (q^(n+1) - q^(-n-1)) / (q - q^(-1)) = q^(-n) (1 - q^(2(n+1)) / (1 - q^2) = q^(-n) sum{k = 0,..,n} q^(2k) = q^(-n) sum_{k = 0,..,n} T(n,k) (1 + q^2)^k. (End) T(n, k) = [x^k] Sum_{j=0..n} (x-1)^j. - Peter Luschny, Jul 09 2019 EXAMPLE 1    0    1    1   -1    1    0    2   -2    1    1   -2    4   -3    1    0    3   -6    7   -4    1    1   -3    9  -13   11   -5    1    0    4  -12   22  -24   16   -6    1    1   -4   16  -34   46  -40   22   -7    1    0    5  -20   50  -80   86  -62   29   -8    1    1   -5   25  -70  130 -166  148  -91   37   -9    1 MAPLE A239473 := proc(n, k)     add(binomial(j, k)*(-1)^(j+k), j=k..n) ; end proc; # R. J. Mathar, Jul 21 2016 MATHEMATICA Table[Sum[(-1)^(j+k)*Binomial[j, k], {j, 0, n}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 06 2018 *) PROG (PARI) for(n=0, 10, for(k=0, n, print1(sum(j=0, n, (-1)^(j+k)*binomial(j, k)), ", "))) \\ G. C. Greubel, Feb 06 2018 (MAGMA) [[(&+[(-1)^(j+k)*Binomial(j, k): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018 (Sage) Trow = lambda n: sum((x-1)^j for j in (0..n)).list() for n in (0..10): print Trow(n) # Peter Luschny, Jul 09 2019 CROSSREFS For column 2: A001057, A004526, A008619, A140106. Column 3: A002620, A087811. Column 4: A002623, A173196. Column 5: A001752. Column 6: A001753. Cf. Bottomley's cross-references in A059260. Embedded in alternating antidiagonals of T are the reversals of arrays A071921 (A225010) and A210220. Cf. A049310. Sequence in context: A238453 A066287 A059260 * A135229 A257543 A081372 Adjacent sequences:  A239470 A239471 A239472 * A239474 A239475 A239476 KEYWORD sign,tabl,easy AUTHOR Tom Copeland, Mar 19 2014 EXTENSIONS Inverse array added by Tom Copeland, Mar 26 2014 STATUS approved

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Last modified October 17 16:51 EDT 2019. Contains 328120 sequences. (Running on oeis4.)