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%I #484 Apr 18 2024 15:48:54
%S 1,1,1,1,3,1,1,5,5,1,1,7,13,7,1,1,9,25,25,9,1,1,11,41,63,41,11,1,1,13,
%T 61,129,129,61,13,1,1,15,85,231,321,231,85,15,1,1,17,113,377,681,681,
%U 377,113,17,1,1,19,145,575,1289,1683,1289,575,145,19,1,1,21,181,833,2241,3653,3653
%N Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
%C In the Formula section, some contributors use T(n,k) = D(n-k, k) (for 0 <= k <= n), which is the triangular version of the square array (D(n,k): n,k >= 0). Conversely, D(n,k) = T(n+k,k) for n,k >= 0. - _Petros Hadjicostas_, Aug 05 2020
%C Also called the tribonacci triangle [Alladi and Hoggatt (1977)]. - _N. J. A. Sloane_, Mar 23 2014
%C D(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), (1,1). - _Joerg Arndt_, Jul 01 2011 [Corrected by _N. J. A. Sloane_, May 30 2020]
%C Or, triangle read by rows of coefficients of polynomials P[n](x) defined by P[0] = 1, P[1] = x+1; for n >= 2, P[n] = (x+1)*P[n-1] + x*P[n-2].
%C D(n, k) is the number of k-matchings of a comb-like graph with n+k teeth. Example: D(1, 3) = 7 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has seven 3-matchings: four triples of three teeth and the three triples {Aa, Bb, CD}, {Aa, Dd, BC}, {Cc, Dd, AB}. Also D(3, 1)=7, the 1-matchings of the same graph being the seven edges: {AB}, {BC}, {CD}, {Aa}, {Bb}, {Cc}, {Dd}. - _Emeric Deutsch_, Jul 01 2002
%C Sum of n-th antidiagonal of the array D is A000129(n+1). - _Reinhard Zumkeller_, Dec 03 2004 [Edited by _Petros Hadjicostas_, Aug 05 2020 so that the counting of antidiagonals of D starts at n = 0. That is, the sum of the terms in the n-th row of the triangles T is A000129(n+1).]
%C The A-sequence for this Riordan type triangle (see one of _Paul Barry_'s comments under Formula) is A112478 and the Z-sequence the trivial: {1, 0, 0, 0, ...}. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. This leads to the recurrence for the triangle given below. - _Wolfdieter Lang_, Jan 21 2008
%C The triangle or chess sums, see A180662 for their definitions, link the Delannoy numbers with twelve different sequences, see the crossrefs. All sums come in pairs due to the symmetrical nature of this triangle. The knight sums Kn14 and Kn15 have been added. It is remarkable that all knight sums are related to the tribonacci numbers, that is, A000073 and A001590, but none of the others. - _Johannes W. Meijer_, Sep 22 2010
%C This sequence, A008288, is jointly generated with A035607 as an array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - _Clark Kimberling_, Mar 09 2012
%C Row n, for n > 0, of _Roger L. Bagula_'s triangle in the Example section shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the numerator of the n-th convergent of the continued fraction [k, k, k, ...], where k = sqrt(x) + 1/sqrt(x); see A230000. - _Clark Kimberling_, Nov 13 2013
%C In an n-dimensional hypercube lattice, D(n,k) gives the number of nodes situated at a Minkowski (Manhattan) distance of k from a given node. In cellular automata theory, the cells at Manhattan distance k are called the von Neumann neighborhood of radius k. For k=1, see A005843. - _Dmitry Zaitsev_, Dec 10 2015
%C These numbers appear as the coefficients of series relating spherical and bispherical harmonics, in the solutions of Laplace's equation in 3D. [Majic 2019, Eq. 22] - _Matt Majic_, Nov 24 2019
%C From _Peter Bala_, Feb 19 2020: (Start)
%C The following remarks assume an offset of 1 in the row and column indices of the triangle.
%C The sequence of row polynomials T(n,x), beginning with T(1,x) = x, T(2,x) = x + x^2, T(3,x) = x + 3*x^2 + x^3, ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd(T(n,x), T(m,x)) = T(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the sequence (T(n,x): n >= 1) is a divisibility sequence in the polynomial ring Z[x]; that is, if n divides m then T(n,x) divides T(m,x) in Z[x].
%C Let S(x) = 1 + 2*x + 6*x^2 + 22*x^3 + ... denote the o.g.f. for the large Schröder numbers A006318. The power series (x*S(x))^n, n = 2, 3, 4, ..., can be expressed as a linear combination with polynomial coefficients of S(x) and 1: (x*S(x))^n = T(n-1,-x) - T(n,-x)*S(x). The result can be extended to negative integer n if we define T(0,x) = 0 and T(-n,x) = (-1)^(n+1) * T(n,x)/x^n. Cf. A115139.
%C [In the previous two paragraphs, D(n,x) was replaced with T(n,x) because the contributor is referring to the rows of the triangle T(n,k), not the rows of the array D(n,k). - _Petros Hadjicostas_, Aug 05 2020] (End)
%C Named after the French amateur mathematician Henri-Auguste Delannoy (1833-1915). - _Amiram Eldar_, Apr 15 2021
%C D(i,j) = D(j,i). With this and _Dmitry Zaitsev_'s Dec 10 2015 comment, D(i,j) can be considered the number of points at L1 distance <= i in Z^j or the number of points at L1 distance <= j in Z^i from any given point. The rows and columns of D(i,j) are the crystal ball sequences on cubic lattices. See the first example below. The n-th term in the k-th crystal ball sequence can be considered the number of points at distance <= n from any point in a k-dimensional cubic lattice, or the number of points at distance <= k from any point in an n-dimensional cubic lattice. - _Shel Kaphan_, Jan 01 2023 and Jan 07 2023
%C Dimensions of hom spaces Hom(R^{(i)}, R^{(j)}) in the Delannoy category attached to the oligomorphic group of order preserving self-bijections of the real line. - _Noah Snyder_, Mar 22 2023
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Mathematica, 26 (1963), 223-229.
%D G. Picou, Note #2235, L'Intermédiaire des Mathématiciens, 8 (1901), page 281. - _N. J. A. Sloane_, Mar 02 2022
%D D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 28.
%H T. D. Noe, <a href="/A008288/b008288.txt">Table of n, a(n) for n = 0..5150</a>
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%H Shel Kaphan, <a href="/A008288/a008288_3.pdf">Tables of Sequences Related to Delannoy Numbers and Cubic Lattice Coordination Numbers</a>
%H Shel Kaphan, <a href="/A008288/a008288_1.txt">Illustration of a recurrence relation on the Delannoy numbers and their connection with geometry.</a>
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%F D(n, 0) = 1 = D(0, n) for n >= 0; D(n, k) = D(n, k-1) + D(n-1, k-1) + D(n-1, k).
%F Bivariate o.g.f.: Sum_{n >= 0, k >= 0} D(n, k)*x^n*y^k = 1/(1 - x - y - x*y).
%F D(n, k) = Sum_{d = 0..min(n,k)} binomial(k, d)*binomial(n+k-d, k) = Sum_{d=0..min(n,k)} 2^d*binomial(n, d)*binomial(k, d). [Edited by _Petros Hadjicostas_, Aug 05 2020]
%F Seen as a triangle read by rows: T(n, 0) = T(n, n) = 1 for n >= 0 and T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), 0 < k < n and n > 1. - _Reinhard Zumkeller_, Dec 03 2004
%F Read as a number triangle, this is the Riordan array (1/(1-x), x(1+x)/(1-x)) with T(n, k) = Sum_{j=0..n-k} C(n-k, j) * C(k, j) * 2^j. - _Paul Barry_, Jul 18 2005
%F T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k). - _Paul Barry_, May 21 2006
%F Let y^k(n) be the number of Khalimsky-continuous functions f from [0,n-1] to Z such that f(0) = 0 and f(n-1) = k. Then y^k(n) = D(i,j) for i = (1/2)*(n-1-k) and j = (1/2)*(n-1+k) where n-1+k belongs to 2Z. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
%F Recurrence for triangle from A-sequence (see the _Wolfdieter Lang_ comment above): T(n,k) = Sum_{j=0..n-k} A112478(j) * T(n-1, k-1+j), n >= 1, k >= 1. [For k > n, the sum is empty, in which case T(n,k) = 0.]
%F From _Peter Bala_, Jul 17 2008: (Start)
%F The n-th row of the square array is the crystal ball sequence for the product lattice A_1 x ... x A_1 (n copies). A035607 is the table of the associated coordination sequences for these lattices.
%F The polynomial p_n(x) := Sum {k = 0..n} 2^k * C(n,k) * C(x,k) = Sum_{k = 0..n} C(n,k) * C(x+k,n), whose values [p_n(0), p_n(1), p_n(2), ... ] give the n-th row of the square array, is the Ehrhart polynomial of the n-dimensional cross polytope (the hyperoctahedron) [Bump et al. (2000), Theorem 6].
%F The first few values are p_0(x) = 1, p_1(x) = 2*x + 1, p_2(x) = 2*x^2 + 2*x + 1 and p_3(x) = (4*x^3 + 6*x^2 + 8*x + 3)/3.
%F The reciprocity law p_n(m) = p_m(n) reflects the symmetry of the table.
%F The polynomial p_n(x) is the unique polynomial solution of the difference equation (x+1)*f(x+1) - x*f(x-1) = (2*n+1)*f(x), normalized so that f(0) = 1.
%F These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane; that is, the polynomials p_n(x-1), n = 1,2,3,..., satisfy a Riemann hypothesis [Bump et al. (2000), Theorem 4]. The o.g.f. for the p_n(x) is (1 + t)^x/(1 - t)^(x + 1) = 1 + (2*x + 1)*t + (2*x^2 + 2*x + 1)*t^2 + ... .
%F The square array of Delannoy numbers has a close connection with the constant log(2). The entries in the n-th row of the array occur in the series acceleration formula log(2) = (1 - 1/2 + 1/3 - ... + (-1)^(n+1)/n) + (-1)^n * Sum_{k>=1} (-1)^(k+1)/(k*D(n,k-1)*D(n,k)). [T(n,k) was replaced with D(n,k) in the formula to agree with the beginning of the paragraph. - _Petros Hadjicostas_, Aug 05 2020]
%F For example, the fourth row of the table (n = 3) gives the series log(2) = 1 - 1/2 + 1/3 - 1/(1*1*7) + 1/(2*7*25) - 1/(3*25*63) + 1/(4*63*129) - ... . See A142979 for further details.
%F Also the main diagonal entries (the central Delannoy numbers) give the series acceleration formula Sum_{n>=1} 1/(n*D(n-1,n-1)*D(n,n)) = (1/2)*log(2), a result due to Burnside. [T(n,n) was replaced here with D(n,n) to agree with the previous paragraphs. - _Petros Hadjicostas_, Aug 05 2020]
%F Similar relations hold between log(2) and the crystal ball sequences of the C_n lattices A142992. For corresponding results for the constants zeta(2) and zeta(3), involving the crystal ball sequences for root lattices of type A_n and A_n x A_n, see A108625 and A143007 respectively. (End)
%F From _Peter Bala_, Oct 28 2008: (Start)
%F Hilbert transform of Pascal's triangle A007318 (see A145905 for the definition of this term).
%F D(n+a,n) = P_n(a,0;3) for all integer a such that a >= -n, where P_n(a,0;x) is the Jacobi polynomial with parameters (a,0) [Hetyei]. The related formula A(n,k) = P_k(0,n-k;3) defines the table of asymmetric Delannoy numbers, essentially A049600. (End)
%F Seen as a triangle read by rows: T(n,k) = (-1)^(n-k) * Hyper2F1([-n+k, k+1], [1], 2) for 0 <= k <= n. - _Peter Luschny_, Aug 02 2014
%F From _Peter Bala_, Jun 25 2015: (Start)
%F O.g.f. for triangle T(n,k): A(z,t) = 1/(1 - (1 + t)*z - t*z^2) = 1 + (1 + t)*z + (1 + 3*t + t^2)*z^2 + (1 + 5*t + 5*t^2 + t^3)*z^3 + ....
%F 1 + z*d/dz(A(z,t))/A(z,t) is the o.g.f. for A102413. (End)
%F E.g.f. for the n-th subdiagonal of T(n,k), n >= 0, equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(2*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 4*x + 4*x^2/2) = 1 + 5*x + 13*x^2/2! + 25*x^3/3! + 41*x^4/4! + 61*x^5/5! + .... - _Peter Bala_, Mar 05 2017 [The n-th subdiagonal of triangle T(n,k) is the n-th row of array D(n,k).]
%F Let a_i(n) be multiplicative with a_i(p^e) = D(i, e), p prime and e >= 0, then Sum_{n > 0} a_i(n)/n^s = (zeta(s))^(2*i+1)/(zeta(2*s))^i for i >= 0. - _Werner Schulte_, Feb 14 2018
%F Seen as a triangle read by rows: T(n,k) = Sum_{i=0..k} binomial(n-i, i) * binomial(n-2*i, k-i) for 0 <= k <= n. - _Werner Schulte_, Jan 09 2019
%F Univariate generating function: Sum_{k >= 0} D(n,k)*z^k = (1 + z)^n/(1 - z)^(n+1). [Dziemianczuk (2013), Eq. 5.3] - _Matt Majic_, Nov 24 2019
%F (n+1)*D(n+1,k) = (2*k+1)*D(n,k) + n*D(n-1,k). [Majic (2019), Eq. 22] - _Matt Majic_, Nov 24 2019
%F For i, j >= 1, D(i,j) = D(i,j-1) + 2*Sum_{k=0..i-1} D(k,j-1), or, because D(i,j) = D(j,i), D(i,j) = D(i-1,j) + 2*Sum_{k=0..j-1} D(i-1,k). - _Shel Kaphan_, Jan 01 2023
%F Sum_{k=0..n} T(n,k)^2 = A026933(n). - _R. J. Mathar_, Nov 07 2023
%e The square array D(i,j) (i >= 0, j >= 0) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... = A000012
%e 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ... = A005408
%e 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, ... = A001844
%e 1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, ... = A001845
%e 1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, ... = A001846
%e ...
%e For D(2,5) = 61, which is seen above in the row labeled A001844, we calculate the sum (9 + 11 + 41) of the 3 nearest terms above and/or to the left. - _Peter Munn_, Jan 01 2023
%e D(2,5) = 61 can also be obtained from the row labeled A005408 using a recurrence mentioned in the formula section: D(2,5) = D(1,5) + 2*Sum_{k=0..4} D(1,k), so D(2,5) = 11 + 2*(1+3+5+7+9) = 11 + 2*25. - _Shel Kaphan_, Jan 01 2023
%e As a triangular array (on its side) this begins:
%e 0, 0, 0, 0, 1, 0, 11, 0, ...
%e 0, 0, 0, 1, 0, 9, 0, 61, ...
%e 0, 0, 1, 0, 7, 0, 41, 0, ...
%e 0, 1, 0, 5, 0, 25, 0, 129, ...
%e 1, 0, 3, 0, 13, 0, 63, 0, ...
%e 0, 1, 0, 5, 0, 25, 0, 129, ...
%e 0, 0, 1, 0, 7, 0, 41, 0, ...
%e 0, 0, 0, 1, 0, 9, 0, 61, ...
%e 0, 0, 0, 0, 1, 0, 11, 0, ...
%e [Edited by _Shel Kaphan_, Jan 01 2023]
%e From _Roger L. Bagula_, Dec 09 2008: (Start)
%e As a triangle T(n,k) (with rows n >= 0 and columns k = 0..n), this begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 5, 5, 1;
%e 1, 7, 13, 7, 1;
%e 1, 9, 25, 25, 9, 1;
%e 1, 11, 41, 63, 41, 11, 1;
%e 1, 13, 61, 129, 129, 61, 13, 1;
%e 1, 15, 85, 231, 321, 231, 85, 15, 1;
%e 1, 17, 113, 377, 681, 681, 377, 113, 17, 1;
%e 1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1;
%e ... (End)
%e Triangle T(n,k) recurrence: 63 = T(6,3) = 25 + 13 + 25 = T(5,2) + T(4,2) + T(5,3).
%e Triangle T(n,k) recurrence with A-sequence A112478: 63 = T(6,3) = 1*25 + 2*25 - 2*9 + 6*1 (T entries from row n = 5 only). [Here the formula T(n,k) = Sum_{j=0..n-k} A112478(j) * T(n-1, k-1+j) is used with n = 6 and k = 3; i.e., T(6,3) = Sum_{j=0..3} A111478(j) * T(5, 2+j). - _Petros Hadjicostas_, Aug 05 2020]
%e From _Philippe Deléham_, Mar 29 2012: (Start)
%e Subtriangle of the triangle given by (1, 0, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 3, 1, 0;
%e 1, 5, 5, 1, 0;
%e 1, 7, 13, 7, 1, 0;
%e 1, 9, 25, 25, 9, 1, 0;
%e 1, 11, 41, 63, 41, 11, 1, 0;
%e ...
%e Subtriangle of the triangle given by (0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 3, 1;
%e 0, 1, 5, 5, 1;
%e 0, 1, 7, 13, 7, 1;
%e 0, 1, 9, 25, 25, 9, 1;
%e 0, 1, 11, 41, 63, 41, 11, 1;
%e ... (End)
%p A008288 := proc(n, k) option remember; if k = 0 then 1 elif n=k then 1 else procname(n-1, k-1) + procname(n-2, k-1) + procname(n-1, k) fi; end: seq(seq(A008288(n,k),k=0..n), n=0..10); # triangular indices n and k
%p P[0]:=1; P[1]:=x+1; for n from 2 to 12 do P[n]:=expand((x+1)*P[n-1]+x*P[n-2]); lprint(P[n]); lprint(seriestolist(series(P[n],x,200))); od:
%t (* Next, A008288 jointly generated with A035607 *)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x];
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A008288 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A035607 *)
%t (* _Clark Kimberling_, Mar 09 2012 *)
%t d[n_, k_] := Binomial[n+k, k]*Hypergeometric2F1[-k, -n, -n-k, -1]; A008288 = Flatten[Table[d[n-k, k], {n, 0, 12}, {k, 0, n}]] (* _Jean-François Alcover_, Apr 05 2012, after 3rd formula *)
%o (Haskell)
%o a008288 n k = a008288_tabl !! n !! k
%o a008288_row n = a008288_tabl !! n
%o a008288_tabl = map fst $ iterate
%o (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
%o zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
%o -- _Reinhard Zumkeller_, Jul 21 2013
%o (Sage)
%o for k in range(8):
%o a = lambda n: hypergeometric([-n, -k], [1], 2)
%o print([simplify(a(n)) for n in range(11)]) # _Peter Luschny_, Nov 19 2014
%o (Python)
%o from functools import cache
%o @cache
%o def delannoy_row(n: int) -> list[int]:
%o if n == 0: return [1]
%o if n == 1: return [1, 1]
%o rov = delannoy_row(n - 2)
%o row = delannoy_row(n - 1) + [1]
%o for k in range(n - 1, 0, -1):
%o row[k] += row[k - 1] + rov[k - 1]
%o return row
%o for n in range(10): print(delannoy_row(n)) # _Peter Luschny_, Jul 30 2023
%Y Sums of antidiagonals: A000129 (Pell numbers).
%Y Main diagonal: A001850 (central Delannoy numbers), which has further information and references.
%Y A002002, A026002, and A190666 are +-k-diagonals for k=1, 2, 3 resp. - _Shel Kaphan_, Jan 01 2023
%Y Rows 0..10: A000012, A005408, A001844, A001845, A001846, A001847, A001848, A001849, A008417, A008419, A008421.
%Y See also A027618.
%Y Cf. A059446.
%Y Has same main diagonal as A064861. Different from A100936.
%Y Cf. A101164, A101167, A128966, A131887, A131935.
%Y Cf. A035607, A108625, A142979, A142992, A143007.
%Y Read mod small primes: A211312, A211313, A211314, A211315.
%Y Triangle sums (see the comments): A000129 (Row1); A056594 (Row2); A000073 (Kn11 & Kn21); A089068 (Kn12 & Kn22); A180668 (Kn13 & Kn23); A180669 (Kn14 & Kn24); A180670 (Kn15 & Kn25); A099463 (Kn3 & Kn4); A116404 (Fi1 & Fi2); A006498 (Ca1 & Ca2); A006498(3*n) (Ca3 & Ca4); A079972 (Gi1 & Gi2); A079972(4*n) (Gi3 & Gi4); A079973(3*n) (Ze1 & Ze2); A079973(2*n) (Ze3 & Ze4).
%Y Cf. A102413, A128966. (D(n,1)) = A005843. Cf. A115139.
%K nonn,tabl,nice,easy,changed
%O 0,5
%A _N. J. A. Sloane_
%E Expanded description from _Clark Kimberling_, Jun 15 1997
%E Additional references from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28 2001
%E Changed the notation to make the formulas more precise. - _N. J. A. Sloane_, Jul 01 2002
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