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A003506
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Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1<=k<=n.
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41
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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, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 11, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 11
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Array 1/Beta(n,m) read by antidiagonals. - Michael Somos Feb 05 2004
a(n,3) = A027480(n-2); a(n,4) = A033488(n-3). - Ross La Haye (rlahaye(AT)new.rr.com), Feb 13 2004
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 (rlahaye(AT)new.rr.com), Dec 31 2006
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 anti-diagonals). - Thomas Wieder (thomas.wieder(AT)t-online.de), Aug 06 2006
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 DELEHAM (kolotoko(AT)wanadoo.fr), Oct 07 2007
This sequence * [1/1, 1/2, 1/3,...] = (1, 3, 7, 15, 31,...) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 14 2007
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 (qntmpkt(AT)yahoo.com), Dec 27 2007
T(n,5)=T(n,n-4)=A174002(n-4) for n>4; T(2*n,n)=T(2*n,n+1)=A005430(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 05 2010]
From Paul Curtz Jun 03 2011. (Start)
Consider
1, 1/2, 1/3, 1/4, 1/5,
-1/2, -1/6, -1/12, -1/20, -1/30,
1/3, 1/12, 1/30, 1/60, 1/105,
-1/4, -1/20, -1/60, -1/140, -1/280,
1/5, 1/30, 1/105, 1/280, 1/630.
This is an eigensequence (the inverse binomial transform is the sequence signed) of second kind:the main diagonal is the double of the first upper diagonal.
Note that 2,12,60,=A005430(n+1), Apery numbers =2*A002457(n). (End)
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, see 130.
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.
G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.
H. J. Brothers, Pascal's Prism: Supplementary Material, http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
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. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.
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LINKS
| Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle
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FORMULA
| a(n, 1)=1/n; a(n, k)=a(n-1, k-1)-a(n, k-1) for k>1.
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 (se16(AT)btinternet.com), Jul 22 2002
G.f.: x*y/(1-x-y*x)^2. E.g.f: x*y*exp(x+x*y). - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 01 2003
T(n,k) = n*binomial(n-1,k-1)= n*A007318(n-1,k-1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 04 2006
Binomial transform of A128064(unsigned). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 29 2007
t(n,m)=Gamma[n]/(Gamma[n - m]*Gamma[m]. - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Sep 14 2008
f[s,n]=Integrate[Exp[ -s*x]*x^n,{x,0,Infinity}]=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 (rlbagulatftn(AT)yahoo.com), Sep 14 2008
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EXAMPLE
| 1/1; 1/2, 1/2; 1/3, 1/6, 1/3; 1/4, 1/12, 1/12, 1/4; 1/5, 1/20, 1/30, 1/20, 1/5; ...
Triangle begins:
{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, 42, 7},
{8, 56, 168, 280, 280, 168, 56, 8},
{9, 72, 252, 504, 630, 504, 252, 72, 9},
{10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10},
{11, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 11}
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MAPLE
| with(combstruct):for n from 0 to 11 do seq(m*count(Combination(n), size=m), m = 1 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 09 2008
A003506 := (n, k) -> k*binomial(n, k):
seq(print(seq(A003506(n, k), k=1..n)), n=1..7); #Peter Luschny, May 27 2011
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MATHEMATICA
| 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[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 (rlbagulatftn(AT)yahoo.com), Sep 14 2008
Table[k*Binomial[n, k], {n, 1, 7}, {k, 1, n}] - Peter Luschny, May 27 2011
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PROG
| (PARI) A(i, j)=if(i<1|j<1, 0, 1/subst(intformal(x^(i-1)*(1-x)^(j-1)), x, 1))
(PARI) A(i, j)=if(i<1|j<1, 0, 1/sum(k=0, i-1, (-1)^k*binomial(i-1, k)/(j+k)))
(PARI) {T(n, k) = (n + 1 - k) * binomial( n, k - 1)} /* Michael Somos, Feb 06 2011 */
(Haskell)
a003506 n k = a003506_tabl !! (n-1) !! (n-1)
a003506_row n = map (* (fromIntegral n)) $ a007318_row (n - 1)
a003506_list = concat a003506_tabl
a003506_tabl = map a003506_row [1..]
-- Reinhard Zumkeller, Nov 17 2011
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CROSSREFS
| Cf. A007622, A128064, A164555/A027642.
Cf. A094305, A121547, A121306, A119800, A002378, A007318.
Row sums are in A001787. Central column is A002457. Half-diagonal is in A090816.
Sequence in context: A128228 A190098 A125102 * A047662 A183474 A075196
Adjacent sequences: A003503 A003504 A003505 * A003507 A003508 A003509
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KEYWORD
| tabl,nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 07 2007
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