A121306 -id:A121306 - cp's OEIS frontend

A003506 Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.

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

Offset: 1

N. J. A. Sloane

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, 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, 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 antidiagonals). - Thomas Wieder, 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 Deléham, Oct 07 2007
This sequence * [1/1, 1/2, 1/3, ...] = (1, 3, 7, 15, 31, ...). - Gary W. Adamson, 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, Dec 27 2007
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 autosequence (the inverse binomial transform is the sequence signed) of the second kind: the main diagonal is 2 times the first upper diagonal.
Note that 2, 12, 60, ... = A005430(n+1), Apery numbers = 2*A002457(n). (End)
From Louis Conover (for the 9th grade G1c mathematics class at the Chengdu Confucius International School), Mar 02 2015: (Start)
The i-th order differences of n^-1 appear in the (i+1)th row.
  1,    1/2,   1/3,   1/4,    1/5,    1/6,    1/7,     1/8, ...
  1/2,  1/6,  1/12,  1/20,   1/30,   1/42,   1/56,    1/72, ...
  1/3, 1/12,  1/30,  1/60,  1/105,  1/168,  1/252,   1/360, ...
  1/4, 1/20,  1/60, 1/140,  1/280,  1/504,  1/840,  1/1320, ...
  1/5, 1/30, 1/105, 1/280,  1/630, 1/1260, 1/2310,  1/3960, ...
  1/6, 1/42, 1/168, 1/504, 1/1260, 1/2772, 1/5544, 1/12012, ...
(End)
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

			The triangle begins:
  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;
  ...
The triangle of denominators 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

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.
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.

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See pp. 8-9.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993.
H. J. Brothers, Pascal's Prism: Supplementary Material.
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Yilmaz Simsek, Construction of a generalization of the Leibnitz numbers and their properties, arXiv:2011.13701 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle

Cf. A007622, A128064, A164555/A027642, A356546.
Cf. A094305, A121547, A121306, A119800, A002378, A007318.
Row sums are in A001787. Central column is A002457. Half-diagonal is in A090816. A116071, A215652.
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
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

a003506 n k = a003506_tabl !! (n-1) !! (n-1)
a003506_row n = a003506_tabl !! (n-1)
a003506_tabl = scanl1 (\xs ys ->
   zipWith (+) (zipWith (+) ([0] ++ xs) (xs ++ [0])) ys) a007318_tabl
a003506_list = concat a003506_tabl
-- Reinhard Zumkeller, Nov 14 2013, Nov 17 2011

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
A003506 := (n,k) -> k*binomial(n,k):
seq(print(seq(A003506(n,k),k=1..n)),n=1..7); # Peter Luschny, May 27 2011

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, Sep 14 2008 *)
Table[k*Binomial[n,k],{n,1,7},{k,1,n}] (* Peter Luschny, May 27 2011 *)
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 *)

A(i,j)=if(i<1||j<1,0,1/subst(intformal(x^(i-1)*(1-x)^(j-1)),x,1))

A(i,j)=if(i<1||j<1,0,1/sum(k=0,i-1,(-1)^k*binomial(i-1,k)/(j+k)))

{T(n, k) = (n + 1 - k) * binomial( n, k - 1)} /* Michael Somos, Feb 06 2011 */

T_row = lambda n: (n*(x+1)^(n-1)).list()
for n in (1..10): print(T_row(n)) # Peter Luschny, Feb 04 2017
# Assuming offset 0:
def A003506(n, k):
    return falling_factorial(n+1,n)//(factorial(k)*factorial(n-k))
for n in range(9): print([A003506(n, k) for k in range(n+1)]) # Peter Luschny, Aug 13 2022

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, Jul 22 2002
From Vladeta Jovovic, Nov 01 2003: (Start)
G.f.: x*y/(1-x-y*x)^2.
E.g.f.: x*y*exp(x+x*y). (End)
T(n,k) = n*binomial(n-1,k-1) = n*A007318(n-1,k-1). - Philippe Deléham, Aug 04 2006
Binomial transform of A128064(unsigned). - Gary W. Adamson, Aug 29 2007
From Roger L. Bagula and Gary W. Adamson, Sep 14 2008: (Start)
t(n,m) = Gamma(n)/(Gamma(n - m)*Gamma(m)).
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. (End)
From Reinhard Zumkeller, Mar 05 2010: (Start)
T(n,5) = T(n,n-4) = A174002(n-4) for n > 4.
T(2*n,n) = T(2*n,n+1) = A005430(n). (End)
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
T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,k+1-i). - Mircea Merca, Apr 11 2012
If we include a main diagonal of zeros so that the array is in the form
  0
  1   0
  2   2   0
  3   6   3   0
  4  12  12   4   0
...
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
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
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

Edited by N. J. A. Sloane, Oct 07 2007

A094305 Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).

Original entry on oeis.org

1, 3, 3, 6, 12, 6, 10, 30, 30, 10, 15, 60, 90, 60, 15, 21, 105, 210, 210, 105, 21, 28, 168, 420, 560, 420, 168, 28, 36, 252, 756, 1260, 1260, 756, 252, 36, 45, 360, 1260, 2520, 3150, 2520, 1260, 360, 45, 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55, 66

Offset: 0

Amarnath Murthy, Apr 29 2004

Sum of all possible sums of k+1 numbers chosen from among the first n+1 numbers. Additive analog of triangle of Stirling numbers of first kind (A008275). - David Wasserman, Oct 04 2007
Third 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
Triangle T(n,k), 0<=k<=n, read by rows given by [3,-1,2/3,-1/6,1/2,0,0,0,0,0,0,...] DELTA [3,-1,2/3,-1/6,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007
T(n,k) is the number of ordered triples of bit strings with n bits and exactly k 1's over all bits in the triple.  For example for n=1 we have (0,e,e),(e,0,e),(e,e,0),(1,e,e),(e,1,e),(e,e,1) where e is the empty string. - Geoffrey Critzer, Apr 06 2013
T(n,k) = A000217(n+1) * A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Jul 30 2013

			Triangle begins:
  1
  3 3
  6 12 6
  10 30 30 10
  15 60 90 60 15
  21 105 210 210 105 21
  ...
The n-th row is the product of the n-th triangular number and the n-th row of Pascal's triangle. The fifth row is (15,60,90,60,15) or 15*{1,4,6,4,1}.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 152.

Reinhard Zumkeller, Rows n = 0..100 of table, flattened
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

For a closely related array that also includes a row and column of zeros see A129533.
Columns include A000217. Row sums are A001788. Cf. A094306.
Cf. A003506, A121547, A121306, A119800, A000217, A007318.

a094305 n k = a094305_tabl !! n !! k
a094305_row n = a094305_tabl !! n
a094305_tabl = zipWith (map . (*)) (tail a000217_list) a007318_tabl
-- Reinhard Zumkeller, Jul 30 2013

A094305:= proc(n,k) (n+1)*(n+2)/2 * binomial(n,k); end;

nn=10; f[list_]:=Select[list,#>0&];a=1/(1-x-y x); Map[f,CoefficientList[Series[a^3,{x,0,nn}],{x,y}]]//Grid
(* Geoffrey Critzer, Apr 06 2013 *)
Flatten[Table[((n+1)(n+2))/2 Binomial[n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Aug 31 2014 *)

T(n,k) = Sum_{i=1..k+1} (-1)^(i+1)*i^2*binomial(n+2,k+i+1)*binomial(n+2,k-i+1). - Mircea Merca, Apr 05 2012

O.g.f.: 1/(1 - x - y*x)^3. - Geoffrey Critzer, Apr 06 2013

Edited by Ralf Stephan, Feb 04 2005
Further comments from David Wasserman, Oct 04 2007
Further editing by N. J. A. Sloane, Oct 07 2007

A121547 Fourth 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) for which the first slice is Pascal's triangle (slice read by antidiagonals).

Original entry on oeis.org

0, 0, 1, 0, 4, 4, 0, 10, 20, 10, 0, 20, 60, 60, 20, 0, 35, 140, 210, 140, 35, 0, 56, 280, 560, 560, 280, 56, 0, 84, 504, 1260, 1680, 1260, 504, 84, 0, 120, 840, 2520, 4200, 4200, 2520, 840, 120, 0, 165, 1320, 4620, 9240, 11550, 9240, 4620, 1320, 165, 0, 220, 1980, 7920, 18480, 27720, 27720, 18480, 7920, 1980, 220

Offset: 0

Thomas Wieder, Aug 06 2006

Essentially the same as triangle A178820 with an additional column of zeros at the left border. - Georg Fischer, Jul 31 2023

			The second row is 1, 4, 10, 20, 35, 56, 84, 120, 165, 220 = A000292, i.e., Tetrahedral (or pyramidal) numbers: binomial(n+2,3) = n(n+1)(n+2)/6 (core).
The third row is 4, 20, 60, 140, 280, 504, 840, 1320, 1980, 2860 = A033488 = n*(n+1)*(n+2)*(n+3)/6.
The main diagonal is 0, 4, 60, 560, 4200, 27720, 168168, 960960, 5250960, 27713400 = {0} U A002803*4.
Triangle starts:
  0
  0, 1
  0, 4, 4
  0, 10, 20, 10
  0, 20, 60, 60, 20
  0, 35, 140, 210, 140, 35
  0, 56, 280, 560, 560, 280, 56
  0, 84, 504, 1260, 1680, 1260, 504, 84

Cf. A000292, A003506, A007318, A094305, A121306, A119800, A178820.

T:=(n, k)->binomial(n+3, 3)*binomial(n, k): seq(print(seq(T(n-1, k-1), k=0..n)), n=0..10); # Georg Fischer, Jul 31 2023

a(m-1,n,o) + a(m,n-1,o) + a(m,n,o-1) with initialization values a(1,0,0) = 1 and a(m<>1=0, n>=0, 0>=o) = 0.

a(55)-a(56) corrected and more terms from Georg Fischer, Jul 31 2023

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A003506 Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.

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A094305 Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).

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A121547 Fourth 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) for which the first slice is Pascal's triangle (slice read by antidiagonals).

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A183153 T(n,k) is the number of order-preserving partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).

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