A003506 -id:A003506 - cp's OEIS frontend

A137754 Numerators (left to right) of Leibniz's harmonic-like triangle.

1, 1, 1, 1, 5, 1, 1, 7, 7, 1, 1, 9, 31, 9, 1, 1, 11, 49, 49, 11, 1, 1, 13, 71, 209, 71, 13, 1, 1, 15, 97, 351, 351, 97, 15, 1, 1, 17, 127, 545, 1471, 545, 127, 17, 1, 1, 19, 161, 799, 2561, 2561, 799, 161, 19, 1, 1, 21, 199, 1121, 4159, 10625, 4159, 1121, 199, 21, 1

Offset: 1

Mohammad K. Azarian, Feb 10 2008

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

Cf. A137752.

Cf. A003506; A007622; A046201; A046204; A046205; A046206; A046208; A046212; A137753.

A137755 Nontrivial numerators (left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

5, 7, 7, 9, 31, 9, 11, 49, 49, 11, 13, 71, 209, 71, 13, 15, 97, 351, 351, 97, 15, 17, 127, 545, 1471, 545, 127, 17, 19, 161, 799, 2561, 2561, 799, 161, 19, 21, 199, 1121, 4159, 10625, 4159, 1121, 199, 21

Offset: 1

Mohammad K. Azarian, Feb 10 2008

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

Cf. A003506; A007622; A046201; A046204; A046205; A046206; A046208; A046212; A137752; A137753; A137754.

A137756 Nontrivial elements in writing the numerator of an element first and then the denominator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the nontrivial elements of A137752.

Original entry on oeis.org

2, 2, 3, 5, 6, 3, 4, 7, 12, 7, 12, 4, 5, 9, 20, 31, 30, 9, 20, 5, 6, 11, 30, 49, 60, 49, 60, 11, 30, 6, 7, 13, 42, 71, 105, 209, 140, 71, 105, 13, 42, 7, 8, 15, 56, 97, 168, 351, 280, 351, 280, 97, 168, 15, 56, 8, 9, 17, 72, 127, 252, 545, 504, 1471

Offset: 1

Mohammad K. Azarian, Feb 10 2008

			1/1; -->
1/2, 1/2; --> 2 2
1/3, 5/6, 1/3; --> 3 5 6 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; A137752; A137753; A137754; A137755.

A137757 Nontrivial elements in writing the denominator of an element first and then the numerator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the nontrivial elements of A137753.

Original entry on oeis.org

2, 2, 3, 6, 5, 3, 4, 12, 7, 12, 7, 4, 5, 20, 9, 30, 31, 20, 9, 5, 6, 30, 11, 60, 49, 60, 49, 30, 11, 6, 7, 42, 13, 105, 71, 140, 209, 105, 71, 42, 13, 7, 8, 56, 15, 168, 97, 280, 351, 280, 351, 168, 97, 56, 15, 8, 9, 72, 17, 252, 127, 504, 545, 630, 1471

Offset: 1

Mohammad K. Azarian, Feb 10 2008

			1/1; -->
1/2, 1/2; --> 2 2
1/3, 5/6, 1/3; --> 3 6 5 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; A137752; A137753; A137754; A137755; A137756.

A137758 Odd elements in writing the denominator of an element first and then the numerator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the odd elements of A137753.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 3, 1, 1, 7, 7, 1, 5, 1, 9, 31, 9, 5, 1, 1, 11, 49, 49, 11, 1, 7, 1, 13, 105, 71, 209, 105, 71, 13, 7, 1, 1, 15, 97, 351, 351, 97, 15, 1, 9, 1, 17, 127, 545, 1471, 545, 127, 17, 9, 1, 1, 19, 161, 799, 2561, 2561, 799, 161, 19, 1

Offset: 1

Mohammad K. Azarian, Feb 10 2008

			1/1; --> 1 1
1/2, 1/2; --> 1 1
1/3, 5/6, 1/3; --> 3 1 5 3 1
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; A137752; A137753; A137754; A137755; A137756; A137757.

A000917 a(n) = (2n+3)!/(n!*(n+2)!).

Original entry on oeis.org

3, 20, 105, 504, 2310, 10296, 45045, 194480, 831402, 3527160, 14872858, 62403600, 260757900, 1085822640, 4508102925, 18668849760, 77138650050, 318107374200, 1309542023790, 5382578744400, 22093039119060, 90567738003600, 370847442355650, 1516927277253024

Offset: 0

N. J. A. Sloane

G.f.: c(x)*(4-c(x))/(1-4*x)^(3/2), c(x) = g.f. for Catalan numbers A000108 (agrees with Hansen, 1975, p. 99, (5.27.9)). Convolution of A038679 with A000984 (central binomial coefficients); also convolution of A038665 with A000302 (powers of 4). - Wolfdieter Lang, Dec 11 1999
Appears as diagonal in A003506. - Zerinvary Lajos, Apr 12 2006
a(n) is the number of double rises in all Grand Dyck paths of semilength n+2. Example: a(0)=3 because in the 6 (=A000984(2)) Grand Dyck paths of semilength 2, namely udud, (uu)dd, uddu, d(uu)d, dudu, dd(uu), we have a total of 3 uu's (shown between parentheses). - Emeric Deutsch, Nov 29 2008

Eldon R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99, (5.27.9).

T. D. Noe, Table of n, a(n) for n = 0..200
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A001622, A001791, A007054, A038665, A038679, A000108, A000984, A000302, A003506, A073010.

1/beta(n, n+2) in A061928.

[(n+1)*Binomial(2*n+3, n+1): n in [0..25]]; // Vincenzo Librandi, Jun 01 2016

a := proc(n) (n+1)*binomial(2*n+3, n+2) end: seq(a(n), n=0..23); # Zerinvary Lajos, Nov 26 2006
seq((n+1)*binomial(2*n+4, n+2)/2, n=0..23); # Zerinvary Lajos, Feb 28 2007

Table[(2*n + 3)!/(n!*(n + 2)!), {n, 0, 25}] (* T. D. Noe, Jun 20 2012 *)

a(n) = (n+1)*binomial(2*n+3, n+1) = (n+1)*A001700(n+1). - Vincenzo Librandi, Jun 01 2016
a(n) = (2*n+3)*A001791(n+1). - R. J. Mathar, Nov 09 2021
D-finite with recurrence +(n+2)*a(n) +10*(-n-1)*a(n-1) +12*(2*n+1)*a(n-2)=0. - R. J. Mathar, Nov 09 2021
D-finite with recurrence n*(n+2)*a(n) -2*(2*n+3)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 09 2021
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 - Pi/(3*sqrt(3)) = 1 - A073010.
Sum_{n>=0} (-1)^n/a(n) = 6*log(phi)/sqrt(5) - 1, where phi is the golden ratio (A001622). (End)

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

A137759 Odd numbers in writing first numerator and then denominator (left to right) of Leibniz's harmonic-like triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 1, 3, 1, 7, 7, 1, 1, 5, 9, 31, 9, 1, 5, 1, 11, 49, 49, 11, 1, 1, 7, 13, 71, 105, 209, 71, 105, 13, 1, 7, 1, 15, 97, 351, 351, 97, 15, 1, 1, 9, 17, 127, 545, 1471, 545, 127, 17, 1, 9, 1, 19, 161, 799, 2561, 2561, 799, 161, 19, 1

Offset: 1

Mohammad K. Azarian, Feb 11 2008

			1/1; --> 1 1
1/2, 1/2; --> 1 1
1/3, 5/6, 1/3; --> 1 3 5 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, A137752, A137753, A137754, A137755, A137756, A137757, A137758.

A090816 a(n) = (3n+1)!/((2n)! * n!).

Original entry on oeis.org

1, 12, 105, 840, 6435, 48048, 352716, 2558160, 18386775, 131231100, 931395465, 6580248480, 46312074900, 324897017760, 2272989850440, 15863901576864, 110487596768703, 768095592509700, 5330949171823275, 36945070220658600, 255702514854135195, 1767643865751234240

Offset: 0

Al Hakanson (hawkuu(AT)excite.com), Feb 11 2004

			a(1) = 4!/(2!*1!) = 24/2 = 12.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Necdet Batir, On certain series involving reciprocals of binomial coefficients, Journal of Classical Analysis, Vol. 2, No. 1 (2013), pp. 1-8.

Cf. A002457, A005809, A016777, A045721, A090957, A090969.

Halfdiagonal of triangle A003506.

[Factorial(3*n+1)/(Factorial(n)*Factorial(2*n)): n in [0..20]]; // G. C. Greubel, Feb 03 2019

a:=n-> binomial(3*n+1,2*n)*(n+1): seq(a(n), n=0..20); # Zerinvary Lajos, Jul 31 2006

f[n_] := 1/Integrate[(x^2 - x^3)^n, {x, 0, 1}]; Table[ f[n], {n, 0, 19}] (* Robert G. Wilson v, Feb 18 2004 *)
Table[1/Beta[2*n+1,n+1], {n,0,20}] (* G. C. Greubel, Feb 03 2019 *)

a(n)=if(n<0,0,(3*n+1)!/(2*n)!/n!) /* Michael Somos, Feb 14 2004 */

a(n)=if(n<0,0,1/subst(intformal((x^2-x^3)^n),x,1)) /* Michael Somos, Feb 14 2004 */

[1/beta(2*n+1,n+1) for n in range(20)] # G. C. Greubel, Feb 03 2019

a(n) = A005809(n) * A016777(n).
a(n) = 1/(Integral_{x=0..1} (x^2 - x^3)^n dx).
G.f.: (((8 + 27*z)*(1/(4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(1/3) + 1/(4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(1/3)) - 3*i*sqrt(3)*sqrt(4 - 27*z)*sqrt(z)*(1/(4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(1/3) - 1/(4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(1/3)))*8^(1/3))/(2*(4 - 27*z)^(3/2)), where i is the imaginary unit. - Karol A. Penson, Feb 06 2024
a(n) = Sum_{k = 0..n} (-1)^(n+k) * (2*n + 2*k + 1)*binomial(2*n+k, k).  This is the particular case m = 1 of the identity Sum_{k = 0..m*n} (-1)^k * (2*n + 2*k + 1) * binomial(2*n+k, k) = (-1)^(m*n) * (m*n + 1) * binomial((m+2)*n+1, 2*n). Cf. A002457 and A306290. - Peter Bala, Nov 02 2024
From Amiram Eldar, Dec 09 2024: (Start)
Sum_{n>=0} 1/a(n) = f(c) = 1.09422712102982285131..., where f(x) = (x*(x-1)/(3*x-1)) * ((3/2)*log(abs(x/(x-1))) + ((3*x-2)/sqrt(3*x^2-4*x)) * (arctan(x/sqrt(3*x^2-4*x)) + arctan((2-x)/sqrt(3*x^2-4*x)))), and c = 2/3 + (1/3)*((25+3*sqrt(69))/2)^(-1/3) + (1/3)*((25+3*sqrt(69))/2)^(1/3).
Sum_{n>=0} (-1)^n/a(n) = f(d) = 0.92513707957813718109..., where f(x) is defined above, and d = 2/3 - (1/3)*((29+3*sqrt(93))/2)^(-1/3) - (1/3)*((29+3*sqrt(93))/2)^(1/3).
Both formulas are from Batir (2013). (End)

New definition from Vladeta Jovovic, Feb 12 2004

A126615 Denominators in a harmonic triangle.

Original entry on oeis.org

1, 2, 2, 2, 6, 3, 2, 6, 12, 4, 2, 6, 12, 20, 5, 2, 6, 12, 20, 30, 6, 2, 6, 12, 20, 30, 42, 7, 2, 6, 12, 20, 30, 42, 56, 8, 2, 6, 12, 20, 30, 42, 56, 72, 9, 2, 6, 12, 20, 30, 42, 56, 72, 90, 10, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 11, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 12, 2, 6

Offset: 1

Gary W. Adamson, Feb 09 2007

The harmonic triangle uses the terms of this sequence as denominators, with numerators = 1: (1/1; 1/2, 1/2; 1/2, 1/6, 1/3; 1/2, 1/6, 1/12, 1/4; 1/2, 1/6, 1/12, 1/10, 1/5; ...). Row sums of the harmonic triangle = 1.

			Triangle T(n,k) begins:
  1;
  2,  2;
  2,  6,  3;
  2,  6, 12,  4;
  2,  6, 12, 20,  5;
  2,  6, 12, 20, 30,  6;
  2,  6, 12, 20, 30, 42,  7;
  ...
1/1 = 1,
1/2 + 1/2 = 1,
1/2 + 1/6 + 1/3 = 1,
1/2 + 1/6 + 1/12 + 1/4 = 1, etc.

Row sums are A006527.

Cf. A000012, A002378, A051340, A127949, A003506.

A126615 := (n,k) -> `if`(n=k, n, 1/Beta(k,2));
seq(print(seq(A126615(n,k), k=1..n)), n=1..9); # Peter Luschny, Jul 27 2014

Flatten@Table[{Array[2PolygonalNumber@#&,n],n+1},{n,0,10}] (* Giorgos Kalogeropoulos, Mar 31 2021 *)

Denominators of the inverse of A127949; numerators = 1. Triangle read by rows, first (n-1) terms of 1*2, 2*3, 3*4, ...; followed by "n".

T(n,k) = k*(k+1) = A002378(k) for k < n; T(n,n) = n. - Andrés Ventas, Mar 26 2021

Gary W. Adamson submitted two different triangles numbered A127899 based on the harmonic numbers. This is the second of them, which I am renumbering as A126615. Unfortunately there were several other entries defined in terms of "A127899" and I may not have guessed which version of A127899 was being referred to. - N. J. A. Sloane, Jan 09 2007

More terms from Philippe Deléham, Dec 17 2008

cp's OEIS Frontend

A137754 Numerators (left to right) of Leibniz's harmonic-like triangle.

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A137755 Nontrivial numerators (left to right) of Leibniz's harmonic-like triangle.

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A137756 Nontrivial elements in writing the numerator of an element first and then the denominator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the nontrivial elements of A137752.

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A137757 Nontrivial elements in writing the denominator of an element first and then the numerator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the nontrivial elements of A137753.

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A137758 Odd elements in writing the denominator of an element first and then the numerator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the odd elements of A137753.

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A000917 a(n) = (2n+3)!/(n!*(n+2)!).

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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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A137759 Odd numbers in writing first numerator and then denominator (left to right) of Leibniz's harmonic-like triangle.

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A090816 a(n) = (3*n+1)!/((2*n)! * n!).

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A090816 a(n) = (3n+1)!/((2n)! * n!).