A003506 -id:A003506 - cp's OEIS frontend

A046212 First numerator and then denominator of central elements of Leibniz's Harmonic Triangle.

1, 1, 1, 6, 1, 30, 1, 140, 1, 630, 1, 2772, 1, 12012, 1, 51480, 1, 218790, 1, 923780, 1, 3879876, 1, 16224936, 1, 67603900, 1, 280816200, 1, 1163381400, 1, 4808643120, 1, 19835652870, 1, 81676217700, 1, 335780006100, 1, 1378465288200, 1

Offset: 1

Mohammad K. Azarian

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

Cf. A002457.

a(2n+1) = A056040(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012

More terms from James Sellers, Dec 13 1999

A046201 Distinct odd numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Original entry on oeis.org

1, 3, 5, 7, 105, 9, 11, 495, 13, 6435, 15, 1365, 15015, 45045, 17, 19, 2907, 21, 101745, 23, 5313, 168245, 1716099, 25, 18386775, 27, 8775, 42181425, 143416845, 29, 593775, 90135045, 882230895, 31, 13485, 849555, 18407025, 181440675

Offset: 1

Mohammad K. Azarian

			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; ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506, A046202.

More terms from James Sellers, Dec 13 1999

A137752 First numerator and then denominator (left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 5, 6, 1, 3, 1, 4, 7, 12, 7, 12, 1, 4, 1, 5, 9, 20, 31, 30, 9, 20, 1, 5, 1, 6, 11, 30, 49, 60, 49, 60, 11, 30, 1, 6, 1, 7, 13, 42, 71, 105, 209, 140, 71, 105, 13, 42, 1, 7, 1, 8, 15, 56, 97, 168, 351, 280, 351, 280, 97, 168

Offset: 1

Mohammad K. Azarian, Feb 10 2008

In this triangle the right-hand edge consists of the reciprocals of the positive integers. A number that is not in this edge is obtained by adding the number diagonally above it to the number to its immediate right. Note that in Leibniz's harmonic triangle we subtract the two numbers to get a number which is not on the right-hand edge.

			1/1;
1/2, 1/2;
1/3, 5/6, 1/3;
1/4, 7/12, 7/12, 1/4;
1/5, 9/20, 31/30, 9/20, 1/5;

Cf. A003506; A007622; A046201; A046204; A046205; A046206; A046208; A046212.

Cf. A137753

A137753 First denominator and then numerator (left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 6, 5, 3, 1, 4, 1, 12, 7, 12, 7, 4, 1, 5, 1, 20, 9, 30, 31, 20, 9, 5, 1, 6, 1, 30, 11, 60, 49, 60, 49, 30, 11, 6, 1, 7, 1, 42, 13, 105, 71, 140, 209, 105, 71, 42, 13, 7, 1, 8, 1, 56, 15, 168, 97, 280, 351, 280, 351, 168, 97

Offset: 1

Mohammad K. Azarian, Feb 10 2008

			1/1; --> 1 1
1/2, 1/2; --> 2 1 2 1
1/3, 5/6, 1/3; --> 3 1 6 5 3 1
1/4, 7/12, 7/12, 1/4; --> 4 1 12 7 12 7 4 1
1/5, 9/20, 31/30, 9/20, 1/5; --> 5 1 20 9 30 31 20 9 5 1

Cf. A003506; A007622; A046201; A046204; A046205; A046206; A046208; A046212; A137752.

Cf. A137752

A046204 Distinct even numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Original entry on oeis.org

2, 6, 4, 12, 20, 30, 60, 42, 140, 8, 56, 168, 280, 72, 252, 504, 630, 10, 90, 360, 840, 1260, 110, 1320, 2310, 2772, 132, 660, 1980, 3960, 5544, 156, 858, 2860, 10296, 12012, 14, 182, 1092, 4004, 10010, 18018, 24024, 210, 5460, 30030, 51480, 16, 240

Offset: 1

Mohammad K. Azarian

			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; ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

More terms from James Sellers, Dec 13 1999

A046206 In Leibniz's Harmonic Triangle, write denominator first and then numerator of each element.

Original entry on oeis.org

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, 1, 6, 1, 30, 1, 60, 1, 60, 1, 30, 1, 6, 1, 7, 1, 42, 1, 105, 1, 140, 1, 105, 1, 42, 1, 7, 1, 8, 1, 56, 1, 168, 1, 280, 1, 280, 1, 168, 1, 56, 1, 8, 1, 9, 1, 72, 1, 252, 1, 504, 1, 630, 1, 504, 1, 252

Offset: 1

Mohammad K. Azarian

			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; ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

More terms from James Sellers, Dec 13 1999

A046208 In Leibniz's Harmonic Triangle, write the numerator first and then the denominator of each element to the right of the central elements.

Original entry on oeis.org

1, 2, 1, 3, 1, 12, 1, 4, 1, 20, 1, 5, 1, 60, 1, 30, 1, 6, 1, 105, 1, 42, 1, 7, 1, 280, 1, 168, 1, 56, 1, 8, 1, 504, 1, 252, 1, 72, 1, 9, 1, 1260, 1, 840, 1, 360, 1, 90, 1, 10, 1, 2310, 1, 1320, 1, 495, 1, 110, 1, 11, 1, 5544, 1, 3960, 1, 1980, 1, 660, 1, 132, 1, 12, 1, 10296, 1

Offset: 1

Mohammad K. Azarian

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

More terms from James Sellers, Dec 13 1999

A090802 Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 8, 12, 12, 6, 16, 32, 48, 48, 24, 32, 80, 160, 240, 240, 120, 64, 192, 480, 960, 1440, 1440, 720, 128, 448, 1344, 3360, 6720, 10080, 10080, 5040, 256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320

Offset: 0

Ross La Haye, Feb 10 2004

Row sums = A010842(n); Row sums from column 1 on = A066534(n) = n*A010842(n-1) = A010842(n) - 2^n.
a(n,k) = n! = k! = A000142(n) for n = k; a(n,n-1) = 2*n! = A052849(n) for n > 1; a(n,n-2) = 2*n! = A052849(n) for n > 2; a(n,n-3) = (4/3)*n! = A082569(n) for n > 3; a(n,n-1)/a(2,1) = n!/2! = A001710(n) for n > 1; a(n,n-2)/ a(3,1) = n!/3! = A001715(n) for n > 2; a(n,n-3)/a(4,1) = n!/4! = A001720(n) for n > 3.
a(2k, k) = A052714(k+1). a(2k-1, k) = A034910(k).
a(n,0) = A000079(n); a(n,1) = A001787(n) = row sums of A003506; a(n,2) = A001815(n) = 2!*A001788(n-1); a(n,3) = A052771(n) = 3!*A001789(n); a(n,4) = A052796(n) = 4!*A003472(n); ceiling[a(n,1) / 2] = A057711(n); a(n,5) = 5!*A054849(n).
In a class of n students, the number of committees (of any size) that contain an ordered k-sized subcommittee is a(n,k). - Ross La Haye, Apr 17 2006
Antidiagonal sums [1,2,5,12,30,76,198,528,1448,4080,...] appear to be binomial transform of A000522 interleaved with itself, i.e., 1,1,2,2,5,5,16,16,65,65,... - Ross La Haye, Sep 09 2006
Let P(A) be the power set of an n-element set A. Then a(n,k) = the number of ways to add k elements of A to each element x of P(A) where the k elements are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007
The derivatives of x^n evaluated at x=2. - T. D. Noe, Apr 21 2011

			{1};
{2, 1};
{4, 4, 2};
{8, 12, 12, 6};
{16, 32, 48, 48, 24};
{32, 80, 160, 240, 240, 120};
{64, 192, 480, 960, 1440, 1440, 720};
{128, 448, 1344, 3360, 6720, 10080, 10080, 5040};
{256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320}
a(5,3) = 240 because P(5,3) = 60, 2^(5-3) = 4 and 60 * 4 = 240.

T. D. Noe, Rows n = 0..100, flattened
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Eric Weisstein, Walk
Eric Weisstein, Boolean Algebra
Eric Weisstein, Hasse Diagram

Cf. A000142, A001710, A001715, A001720, A001787, A001788, A001789, A001815, A003472, A010842, A052771, A052796, A052849, A054849, A057711, A066534, A082569.

Cf. A038207, A007318.

Flatten[Table[n!/(n-k)! * 2^(n-k), {n, 0, 8}, {k, 0, n}]] (* Ross La Haye, Feb 10 2004 *)

a(n, k) = 0 for n < k. a(n, k) = k!*C(n, k)*2^(n-k) = P(n, k)*2^(n-k) = (2n)!!/((n-k)!*2^k) = k!*A038207(n, k) = A068424*2^(n-k) = Sum[C(n, m)*P(n-m, k), {m, 0, n-k}] = Sum[C(n, n-m)*P(n-m, k), {m, 0, n-k}] = n!*Sum[1/(m!*(n-m-k)!), {m, 0, n-k}] = k!*Sum[C(n, m)*C(n-m, k), {m, 0, n-k}] = k!*Sum[C(n, n-m)*C(n-m, k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, n-m-k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, m), {m, 0, n-k}] for n >= k.
a(n, k) = 0 for n < k. a(n, k) = n*a(n-1, k-1) for n >= k >= 1.
E.g.f. (by columns): exp(2x)*x^k.

More terms from Ray Chandler, Feb 26 2004

Entry revised by Ross La Haye, Aug 18 2006

A033488 a(n) = n(n+1)(n+2)*(n+3)/6.

Original entry on oeis.org

0, 4, 20, 60, 140, 280, 504, 840, 1320, 1980, 2860, 4004, 5460, 7280, 9520, 12240, 15504, 19380, 23940, 29260, 35420, 42504, 50600, 59800, 70200, 81900, 95004, 109620, 125860, 143840, 163680, 185504, 209440, 235620, 264180, 295260, 329004, 365560, 405080

Offset: 0

N. J. A. Sloane

With two initial 0, convolution of the oblong numbers (A002378) with the nonnegative even numbers (A005843). - Bruno Berselli, Oct 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..700

1/beta(n, 4) in A061928.
Cf. A000332, A034827, A050534.
Cf. A002378, A005843.
Convolution of the oblong numbers with the odd numbers: A008911.
Fourth column of A003506.

[n*(n+1)*(n+2)*(n+3)/6: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(4*binomial(n+3, 4), n=0..35)]; # Zerinvary Lajos, Nov 24 2006

f[n_]:=n*(n+1)*(n+2)*(n+3)/6; lst={};Do[AppendTo[lst,f[n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
# Binomial[#+3,3]&/@ Range[0,40]  (* Harvey P. Dale, Feb 20 2011 *)

A033488(n):=n*(n+1)*(n+2)*(n+3)/6$ makelist(A033488(n),n,0,20); /* Martin Ettl, Jan 22 2013 */

a(n) = n*C(3+n, 3). - Zerinvary Lajos, Jan 10 2006
G.f.: 4*x/(1-x)^5. - Colin Barker, Mar 01 2012
G.f.: (2*x/(1-x))*W(0), where W(k) = 1 + 1/( 1 - x*(k+2)*(k+4)/( x*(k+2)*(k+4) + (k+1)*(k+2)/W(k+1) ) ); (continued fraction). - Sergei N. Gladkovskii, Aug 24 2013
From Amiram Eldar, Jun 02 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2) - 16/3. (End)
E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/6. - Stefano Spezia, Jul 11 2025

A046205 In Leibniz's Harmonic Triangle, write numerator first and then denominator of each element.

Original entry on oeis.org

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, 1, 6, 1, 30, 1, 60, 1, 60, 1, 30, 1, 6, 1, 7, 1, 42, 1, 105, 1, 140, 1, 105, 1, 42, 1, 7, 1, 8, 1, 56, 1, 168, 1, 280, 1, 280, 1, 168, 1, 56, 1, 8, 1, 9, 1, 72, 1, 252, 1, 504, 1, 630, 1, 504, 1

Offset: 1

Mohammad K. Azarian

			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; ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506, A046206, A046212, A046201, A046204, A046208, A007622, A137752, A137753, A137754.

More terms from Gregory D Johnson (gjohn(AT)iname.com)

Edited by M. F. Hasler, Apr 05 2015

cp's OEIS Frontend

A046212 First numerator and then denominator of central elements of Leibniz's Harmonic Triangle.

Views

Author

Keywords

References

Crossrefs

Formula

Extensions

A046201 Distinct odd numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Views

Author

Keywords

Examples

References

Crossrefs

Extensions

A137752 First numerator and then denominator (left to right) of Leibniz's harmonic-like triangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

A137753 First denominator and then numerator (left to right) of Leibniz's harmonic-like triangle.

Views

Author

Keywords

Examples

Crossrefs

A046204 Distinct even numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Views

Author

Keywords

Examples

References

Crossrefs

Extensions

A046206 In Leibniz's Harmonic Triangle, write denominator first and then numerator of each element.

Views

Author

Keywords

Examples

References

Crossrefs

Extensions

A046208 In Leibniz's Harmonic Triangle, write the numerator first and then the denominator of each element to the right of the central elements.

Views

Author

Keywords

References

Crossrefs

Extensions

A090802 Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A033488 a(n) = n*(n+1)*(n+2)*(n+3)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

Formula

A046205 In Leibniz's Harmonic Triangle, write numerator first and then denominator of each element.

Views

A033488 a(n) = n(n+1)(n+2)*(n+3)/6.