%I #98 Jun 22 2024 16:18:21
%S 1,-1,1,1,-2,1,-1,3,-3,1,1,-4,6,-4,1,-1,5,-10,10,-5,1,1,-6,15,-20,15,
%T -6,1,-1,7,-21,35,-35,21,-7,1,1,-8,28,-56,70,-56,28,-8,1,-1,9,-36,84,
%U -126,126,-84,36,-9,1,1,-10,45,-120,210,-252,210,-120,45,-10,1,-1,11,-55,165,-330,462,-462,330,-165,55,-11,1
%N Triangle read by rows: lower triangular matrix which is inverse to Pascal's triangle (A007318) regarded as a lower triangular matrix.
%C Triangle T(n,k), read by rows, given by [-1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C Coefficients of the polynomials generated by the e.g.f. exp(x*t)*exp(-t). - _Peter Luschny_, Jul 13 2009
%C Riordan array (1/(1+x), x/(1+x)). - _Philippe Deléham_, Nov 29 2009
%C Multiplication of a sequence (written as column vector) by this matrix (to the left) yields the inverse Binomial transform of the sequence. - _M. F. Hasler_, Nov 01 2014
%C From _Tom Copeland_, Nov 16 2016: (Start)
%C This signed Pascal matrix IP and the Pascal matrix P contain the coefficients of a prototypical pair of Appell polynomial sequences that are inverse under umbral composition with e.g.f.s e^((x-1)*t) = e^(-t) e^(xt) = f(t) e^(xt) and e^((x+1)t) = e^t e^(xt) = g(t) e^(xt) and row polynomials q_n(x) = (x-1)^n and p_n(x) = (x+1)^n, respectively. The inverse property for an Appell pair is reflected in IP*P = identity matrix, f(t) = 1/g(t), the umbral relation p_n(q.(x)) = x^n = q_n(p.(x)), and their respective raising operators R_(Ip) = x - h(D) and R_P = x + h(D) differing only in the sign of the differential term (h(D) = 1, in this case). The lowering operator for an Appell sequence is L = D = d/dx with L p_n(x) = n*p_(n-1)(x), and the raising operator is defined by R p_n(x) = p_(n+1)(x).
%C The related signed Pascal matrix M with M(n,k) = (-1)^n IP(n,k) = (-1)^k P(n,k) has the e.g.f. e^((1-x)t) = e^t e^(-xt), and w_n(x) = (1-x)^n is not an Appell sequence, but it is a Sheffer sequence with lowering and raising operators L = -D and R = 1 - x, and M = M^(-1) since w_n(w.(x)) = (1-w.(x))^n = sum_{k = 0,..,n} binomial(n,k) (-1)^k w_k(x) = (1-(1-x))^n = x^n.
%C Umbral composition of a pair of Sheffer polynomial sequences, of which Appell sequences are a special class, is equivalent to the multiplication of their respective coefficient matrices.
%C (End)
%H Reinhard Zumkeller, <a href="/A130595/b130595.txt">Rows n = 0..125 of triangle, flattened</a>
%H Shishuo Fu, Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.
%H Tian-Xiao He and Renzo Sprugnoli, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.021">Sequence characterization of Riordan arrays</a>, Discrete Math. 309 (2009), no. 12, 3962-3974. [_N. J. A. Sloane_, Nov 26 2011]
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Harmonic_number#Calculation">Relation between binomial coefficients and harmonic numbers</a>.
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F T(n,k) = (-1)^(n-k)*binomial(n,k) = (-1)^(n-k)*A007318(n,k).
%F T(n,k) = T(n-1,k-1) - T(n-1,k). - _Philippe Deléham_, Oct 10 2011
%F G.f.: 1/(1+x-x*y). - _R. J. Mathar_, Aug 11 2015 [corrected by _Anders Claesson_, Nov 28 2015]
%F Conjecture from _Dale Gerdemann_, Nov 28 2015:
%F T(n,k) = (n-k+1)*T(n-1,k-1) + (k-1)*T(n-1,k).
%F Proof from _Anders Claesson_, Nov 29 2015:
%F It follows from T(n,k) = T(n-1,k-1) - T(n-1,k) and n*T(n-1,k-1) = k*T(n,k) that: (n-k+1)*T(n-1,k-1) + (k-1)*T(n-1,k) = n*T(n-1,k-1) - (k-1)*T(n-1,k-1) + (k-1)*T(n-1,k) = n*T(n-1,k-1) - (k-1)*(T(n-1,k-1) - T(n-1,k)) = n*T(n-1,k-1) - (k-1)*T(n,k) = n*T(n-1,k-1) - k*T(n,k) + T(n,k) = T(n,k). QED
%F (-1)^(n+1) Sum_{k=1..n} T(n,k)/k = Sum_{k=1..n} 1/k = H(n) where H(n) is the n-th harmonic number. For a proof see link "Relation between binomial coefficients and harmonic numbers". - _Wolfgang Hintze_, Oct 22 2016
%F T(n,k) = binomial(-1-k,n-k). - _Robert A. Russell_, Jan 16 2020
%F From _G. C. Greubel_, Jun 22 2024: (Start)
%F T(n, n-k) = (-1)^n*T(n, k).
%F Sum_{k=0..n} T(n, k) = A000007(n).
%F Sum_{k=0..n} (-1)^k*T(n, k) = A122803(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A039834(n+1).
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A049347(n).
%F Sum_{k=0..n} k*T(n, k) = A063524(n).
%F Sum_{k=0..n} (-1)^k*k*T(n, k) = A085750(n+1).
%F Sum_{k=0..n} (k+1)*T(n, k) = A019590(n). (End)
%e Triangle begins with T(0,0):
%e 1;
%e -1, 1;
%e 1, -2, 1;
%e -1, 3, -3, 1;
%e 1, -4, 6, -4, 1;
%e -1, 5, -10, 10, -5, 1;
%e 1, -6, 15, -20, 15, -6, 1;
%e -1, 7, -21, 35, -35, 21, -7, 1;
%e 1, -8, 28, -56, 70, -56, 28, -8, 1;
%e -1, 9, -36, 84, -126, 126, -84, 36, -9, 1;
%e ...
%e As polynomials:
%e + 1;
%e - 1 + 1 x;
%e + 1 - 2 x + 1 x^2;
%e - 1 + 3 x - 3 x^2 + 1 x^3;
%e + 1 - 4 x + 6 x^2 - 4 x^3 + 1 x^4;
%p A130595 := proc(n,k)
%p (-1)^(n+k)*binomial(n,k) ;
%p end proc: # _R. J. Mathar_, Feb 13 2013
%t nmax = 11; t[n_, k_] := (-1)^(n-k)*Binomial[n, k]; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}] ] (* _Jean-François Alcover_, Dec 01 2011 *)
%t Table[Binomial[-1-k, n-k],{n,0,11},{k,0,n}]//Flatten (* _Robert A. Russell_, Jan 16 2020 *)
%o (Haskell)
%o a130595 n = a130595_list !! n
%o a130595_list = concat $ iterate ([-1,1] *) [1]
%o instance Num a => Num [a] where
%o fromInteger k = [fromInteger k]
%o (p:ps) + (q:qs) = p + q : ps + qs
%o ps + qs = ps ++ qs
%o (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
%o _ * _ = []
%o -- _Reinhard Zumkeller_, Apr 02 2011
%o (Haskell)
%o a130595 n k = a130595_tabl !! n !! k
%o a130595_row n = a130595_tabl !! n
%o a130595_tabl = iterate (\row -> zipWith (-) ([0] ++ row) (row ++ [0])) [1]
%o -- _Reinhard Zumkeller_, Apr 13 2013
%o (PARI) A130595(n,k)=(-1)^(n+k)*binomial(n,k) \\ _M. F. Hasler_, Nov 01 2014
%o (Magma) [(-1)^(n+k)*Binomial(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 22 2024
%o (SageMath) flatten([[(-1)^(n+k)*binomial(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Jun 22 2024
%Y Cf. A007318, A084938.
%Y Sums include: A000007 (row sums), A019590, A039834 (diagonal sums), A049347 (alternating sign diagonal sums), A063524, A085750, A122803 (alternating sign sums).
%K sign,nice,tabl
%O 0,5
%A _Philippe Deléham_, Jun 17 2007
%E Edited by _N. J. A. Sloane_, Nov 27 2011