%I #146 Dec 03 2022 09:30:22
%S 1,2,2,3,6,3,4,12,12,4,5,20,30,20,5,6,30,60,60,30,6,7,42,105,140,105,
%T 42,7,8,56,168,280,280,168,56,8,9,72,252,504,630,504,252,72,9,10,90,
%U 360,840,1260,1260,840,360,90,10,11,110,495,1320,2310,2772,2310,1320,495,110,11
%N Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
%C Array 1/Beta(n,m) read by antidiagonals. - _Michael Somos_, Feb 05 2004
%C a(n,3) = A027480(n-2); a(n,4) = A033488(n-3). - _Ross La Haye_, Feb 13 2004
%C a(n,k) = total size of all of the elements of the family of k-size subsets of an n-element set. For example, a 2-element set, say, {1,2}, has 3 families of k-size subsets: one with 1 0-size element, one with 2 1-size elements and one with 1 2-size element; respectively, {{}}, {{1},{2}}, {{1,2}}. - _Ross La Haye_, Dec 31 2006
%C Second slice along the 1-2-plane in the cube a(m,n,o) = a(m-1,n,o) + a(m,n-1,o) + a(m,n,o-1) with a(1,0,0)=1 and a(m<>1=0,n>=0,0>=o)=0, for which the first slice is Pascal's triangle (slice read by antidiagonals). - _Thomas Wieder_, Aug 06 2006
%C Triangle, read by rows, given by [2,-1/2,1/2,0,0,0,0,0,0,...] DELTA [2,-1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 07 2007
%C This sequence * [1/1, 1/2, 1/3, ...] = (1, 3, 7, 15, 31, ...). - _Gary W. Adamson_, Nov 14 2007
%C n-th row = coefficients of first derivative of corresponding Pascal's triangle row. Example: x^4 + 4x^3 + 6x^2 + 4x + 1 becomes (4, 12, 12, 4). - _Gary W. Adamson_, Dec 27 2007
%C From _Paul Curtz_, Jun 03 2011: (Start)
%C Consider
%C 1 1/2 1/3 1/4 1/5
%C -1/2 -1/6 -1/12 -1/20 -1/30
%C 1/3 1/12 1/30 1/60 1/105
%C -1/4 -1/20 -1/60 -1/140 -1/280
%C 1/5 1/30 1/105 1/280 1/630
%C This is an autosequence (the inverse binomial transform is the sequence signed) of the second kind: the main diagonal is 2 times the first upper diagonal.
%C Note that 2, 12, 60, ... = A005430(n+1), Apery numbers = 2*A002457(n). (End)
%C From _Louis Conover_ (for the 9th grade G1c mathematics class at the Chengdu Confucius International School), Mar 02 2015: (Start)
%C The i-th order differences of n^-1 appear in the (i+1)th row.
%C 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, ...
%C 1/2, 1/6, 1/12, 1/20, 1/30, 1/42, 1/56, 1/72, ...
%C 1/3, 1/12, 1/30, 1/60, 1/105, 1/168, 1/252, 1/360, ...
%C 1/4, 1/20, 1/60, 1/140, 1/280, 1/504, 1/840, 1/1320, ...
%C 1/5, 1/30, 1/105, 1/280, 1/630, 1/1260, 1/2310, 1/3960, ...
%C 1/6, 1/42, 1/168, 1/504, 1/1260, 1/2772, 1/5544, 1/12012, ...
%C (End)
%C T(n,k) is the number of edges of distance k from a fixed vertex in the n-dimensional hypercube. - _Simon Burton_, Nov 04 2022
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, see 130.
%D B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 38.
%D G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
%D M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. See p. 152.
%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.
%H Reinhard Zumkeller, <a href="/A003506/b003506.txt">Rows n = 1..120 of triangle, flattened</a>
%H H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>.
%H 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 Yilmaz Simsek, <a href="https://arxiv.org/abs/2011.13701">Construction of a generalization of the Leibnitz numbers and their properties</a>, arXiv:2011.13701 [math.NT], 2020.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html">Leibniz Harmonic Triangle</a>
%F a(n, 1) = 1/n; a(n, k) = a(n-1, k-1) - a(n, k-1) for k > 1.
%F Considering the integer values (rather than unit fractions): a(n, k) = k*C(n, k) = n*C(n-1, k-1) = a(n, k-1)*a(n-1, k-1)/(a(n, k-1) - a(n-1, k-1)) = a(n-1, k) + a(n-1, k-1)*k/(k-1) = (a(n-1, k) + a(n-1, k-1))*n/(n-1) = k*A007318(n, k) = n*A007318(n-1, k-1). Row sums of integers are n*2^(n-1) = A001787(n); row sums of the unit fractions are A003149(n-1)/A000142(n). - _Henry Bottomley_, Jul 22 2002
%F From _Vladeta Jovovic_, Nov 01 2003: (Start)
%F G.f.: x*y/(1-x-y*x)^2.
%F E.g.f.: x*y*exp(x+x*y). (End)
%F T(n,k) = n*binomial(n-1,k-1) = n*A007318(n-1,k-1). - _Philippe Deléham_, Aug 04 2006
%F Binomial transform of A128064(unsigned). - _Gary W. Adamson_, Aug 29 2007
%F t(n,m) = Gamma(n)/(Gamma(n - m)*Gamma(m). - _Roger L. Bagula_ and _Gary W. Adamson_, Sep 14 2008
%F f(s,n) = Integral_{x=0..oo} exp(-s*x)*x^n dx = Gamma(n)/s^n; t(n,m) = f(s,n)/(f(s,n-m)*f(s,m)) = Gamma(n)/(Gamma(n - m)*Gamma(m); the powers of s cancel out. - _Roger L. Bagula_ and _Gary W. Adamson_, Sep 14 2008
%F From _Reinhard Zumkeller_, Mar 05 2010: (Start)
%F T(n,5) = T(n,n-4) = A174002(n-4) for n > 4.
%F T(2*n,n) = T(2*n,n+1) = A005430(n). (End)
%F T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-2,k-2), T(1,1) = 1 and, for n > 1, T(n,k) = 0 if k <= 1 or if k > n. - _Philippe Deléham_, Mar 17 2012
%F T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,k+1-i). - _Mircea Merca_, Apr 11 2012
%F If we include a main diagonal of zeros so that the array is in the form
%F 0
%F 1 0
%F 2 2 0
%F 3 6 3 0
%F 4 12 12 4 0
%F ...
%F then we obtain the exponential Riordan array [x*exp(x),x], which factors as [x,x]*[exp(x),x] = A132440*A007318. This array is the infinitesimal generator for A116071. A signed version of the array is the infinitesimal generator for A215652. - _Peter Bala_, Sep 14 2012
%F a(n,k) = (n-1)!/((n-k)!(k-1)!) if k > n/2 and a(n,k) = (n-1)!/((n-k-1)!k!) otherwise. [Forms 'core' for Pascal's recurrence; gives common term of RHS of T(n,k) = T(n-1,k-1) + T(n-1,k)]. - _Jon Perry_, Oct 08 2013
%F Assuming offset 0: T(n, k) = FallingFactorial(n + 1, n) / (k! * (n - k)!). The counterpart using the rising factorial is A356546. - _Peter Luschny_, Aug 13 2022
%e The triangle begins:
%e 1;
%e 1/2, 1/2;
%e 1/3, 1/6, 1/3;
%e 1/4, 1/12, 1/12, 1/4;
%e 1/5, 1/20, 1/30, 1/20, 1/5;
%e ...
%e The triangle of denominators begins:
%e 1
%e 2 2
%e 3 6 3
%e 4 12 12 4
%e 5 20 30 20 5
%e 6 30 60 60 30 6
%e 7 42 105 140 105 42 7
%e 8 56 168 280 280 168 56 8
%e 9 72 252 504 630 504 252 72 9
%e 10 90 360 840 1260 1260 840 360 90 10
%e 11 110 495 1320 2310 2772 2310 1320 495 110 11
%p with(combstruct):for n from 0 to 11 do seq(m*count(Combination(n), size=m), m = 1 .. n) od; # _Zerinvary Lajos_, Apr 09 2008
%p A003506 := (n,k) -> k*binomial(n,k):
%p seq(print(seq(A003506(n,k),k=1..n)),n=1..7); # _Peter Luschny_, May 27 2011
%t L[n_, 1] := 1/n; L[n_, m_] := L[n, m] = L[n - 1, m - 1] - L[n, m - 1]; Take[ Flatten[ Table[ 1 / L[n, m], {n, 1, 12}, {m, 1, n}]], 66]
%t t[n_, m_] = Gamma[n]/(Gamma[n - m]*Gamma[m]); Table[Table[t[n, m], {m, 1, n - 1}], {n, 2, 12}]; Flatten[%] (* _Roger L. Bagula_ and _Gary W. Adamson_, Sep 14 2008 *)
%t Table[k*Binomial[n,k],{n,1,7},{k,1,n}] (* _Peter Luschny_, May 27 2011 *)
%t t[n_, k_] := Denominator[n!*k!/(n+k+1)!]; Table[t[n-k, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 28 2013 *)
%o (PARI) A(i,j)=if(i<1||j<1,0,1/subst(intformal(x^(i-1)*(1-x)^(j-1)),x,1))
%o (PARI) A(i,j)=if(i<1||j<1,0,1/sum(k=0,i-1,(-1)^k*binomial(i-1,k)/(j+k)))
%o (PARI) {T(n, k) = (n + 1 - k) * binomial( n, k - 1)} /* _Michael Somos_, Feb 06 2011 */
%o (Haskell)
%o a003506 n k = a003506_tabl !! (n-1) !! (n-1)
%o a003506_row n = a003506_tabl !! (n-1)
%o a003506_tabl = scanl1 (\xs ys ->
%o zipWith (+) (zipWith (+) ([0] ++ xs) (xs ++ [0])) ys) a007318_tabl
%o a003506_list = concat a003506_tabl
%o -- _Reinhard Zumkeller_, Nov 14 2013, Nov 17 2011
%o (SageMath)
%o T_row = lambda n: (n*(x+1)^(n-1)).list()
%o for n in (1..10): print(T_row(n)) # _Peter Luschny_, Feb 04 2017
%o # Assuming offset 0:
%o def A003506(n, k):
%o return falling_factorial(n+1,n)//(factorial(k)*factorial(n-k))
%o for n in range(9): print([A003506(n, k) for k in range(n+1)]) # _Peter Luschny_, Aug 13 2022
%Y Cf. A007622, A128064, A164555/A027642, A356546.
%Y Cf. A094305, A121547, A121306, A119800, A002378, A007318.
%Y Row sums are in A001787. Central column is A002457. Half-diagonal is in A090816. A116071, A215652.
%Y Denominators of i-th order differences of n^-1 are given in: (1st) A002378, (2nd) A027480, (3rd) A033488, (4th) A174002, (5th) A253946. - _Louis Conover_, Mar 02 2015
%Y Columns k >= 1 (offset 1): A000027, A002378, A027480, A033488, A174002, A253946(n+4), ..., with sum of reciprocals: infinity, 1, 1/2, 1/3, 1/4, 1/5, ..., respectively. - _Wolfdieter Lang_, Jul 20 2022
%K tabl,nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Oct 07 2007