A003506 -id:A003506 - cp's OEIS frontend

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 5, 20, 30, 20, 5, 10, 30, 30, 10, 10, 20, 10, 5, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1

Offset: 0

Harlan J. Brothers, Jun 16 2010

P_h = level h of Pascal's prism where P_1 = Pascal's triangle (A007318) and P_2 = denominators of Leibniz harmonic triangle (A003506). A sequence of length k through P is defined by P for n = {1, 2, 3, ..., k}.

			Prism begins (levels 1-4):
1
1 1
1 2 1
1 3 3 1
1
2 2
3 6 3
4 12 12 4
1
3 3
6 12 6
10 30 30 10
1
4 4
10 20 10
20 60 60 20

H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Level 1 = A007318.
Level 2 = A003506.
Level 3 = A094305.
Level 4 = A178820.
Level 5 = A178821.
Level 6 = A178822.
Sums of shallow diagonals for each level correspond to rows of square A037027.
Contains A109649 and A046816.
P = A000984.
P = A006480.
P = A000897.
P<3n-2, 3n-2, n> = A113424.

end = 5; Column/@Table[Multinomial[h, i-j, j], {h, 0, end}, {i, 0, end}, {j, 0, i}]

a_(h, i, j) = (h+i-2; h-1, i-j, j-1), h >= 1, i >= 1, 1 <= j <= i.
Recurrence:
For P_h, element a is given by: a_(1, 1) = 1; a_(i, j) = ((i+h-2)/(i-1)) (a_(i-1, j) + a_(i-1, j-1)).

Keyword tabf by Michel Marcus, Oct 22 2017

A241519 Denominators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

Original entry on oeis.org

1, 2, 2, 12, 3, 15, 60, 840, 105, 630, 630, 13860, 6930, 180180, 360360, 144144, 9009, 306306, 306306, 11639628, 14549535, 14549535, 58198140, 2677114440, 334639305, 3346393050

Offset: 0

Paul Curtz, Apr 24 2014

nonn

Generally, 2*b(n) = b(n-1) + f(n). See, for f(n)=n, A000337(n)/2^n.
a(0)=1. b(n) is mentioned in A241269.
Difference table of b(n):
     0,   1/2,   1/2,  5/12,    1/3,   4/15, ...
   1/2,     0, -1/12, -1/12,  -1/15,  -1/20, ...
  -1/2, -1/12,     0,  1/60,   1/60, 11/840, ...
  5/12,  1/12,  1/60,     0, -1/280, -1/280, ...
etc.
b(n) is mentioned in A241269 as an autosequence of the first kind.
The denominators of the first two upper diagonals are the positive Apéry numbers, A005430(n+1). Compare to the array in A003506.
Numerators: 0, 1, 1, 5, 1, 4, 13, 151, 16, 83, 73, 1433, 647, 15341, ... .

			0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ...
b(1) = (0+1)/2, hence a(1)=2.
b(2) = (1/2+1/2)/2 = 1/2, hence a(2)=2.
b(3) = (1/2+1/3)/2 = 5/12, hence a(3)=12.

Cf. A086466.

Cf. A242376 (numerators).

b[0] = 0; b[n_] := b[n] = 1/2*(b[n-1] + 1/n); Table[b[n] // Denominator, {n, 0, 25}] (* Jean-François Alcover, Apr 25 2014 *)
Table[-Re[LerchPhi[2, 1, n + 1]], {n, 0, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
-Re[LerchPhi[2, 1, Range[20]]] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)
RecurrenceTable[{b[n] == b[n - 1]/2 + 1/(2 n), b[0] == 0}, b[n], {n, 20}] // Denominator (* Eric W. Weisstein, Dec 11 2017 *)

b(n) = -Re(Phi(2, 1, n + 1)) where Phi denotes the Lerch transcendent. - Eric W. Weisstein, Dec 11 2017

Extension, after a(13), from Jean-François Alcover, Apr 24 2014

A356546 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

Original entry on oeis.org

1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870

Offset: 0

Peter Luschny, Aug 12 2022

The counterpart using the falling factorial is Leibniz's Harmonic Triangle A003506.

			Triangle T(n, k) begins:
[0]     1;
[1]     2,      2;
[2]     6,     12,      6;
[3]    20,     60,     60,     20;
[4]    70,    280,    420,    280,     70;
[5]   252,   1260,   2520,   2520,   1260,    252;
[6]   924,   5544,  13860,  18480,  13860,   5544,    924;
[7]  3432,  24024,  72072, 120120, 120120,  72072,  24024,   3432;
[8] 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870;

cf. A000984, A059304 (row sums, see also A343842), A265609 (rising factorial).

Cf. A003506, A173018 (Eulerian numbers), A000108, A000897 (central terms).

A356546 := (n, k) -> pochhammer(n+1, n)/(k!*(n-k)!):
for n from 0 to 8 do seq(A356546(n, k), k=0..n) od;

T[ n_, k_] := Binomial[2*n, n] * Binomial[n, k]; (* Michael Somos, Aug 18 2022 *)

{T(n, k) = binomial(2*n, n) * binomial(n, k)}; /* Michael Somos, Aug 18 2022 */

def A356546(n, k):
    return rising_factorial(n+1,n) // (factorial(k) * factorial(n-k))
for n in range(9): print([A356546(n, k) for k in range(n+1)])

Bernoulli(n) / Catalan(n) = Sum_{k=0..n} (-1)^k*A173018(n, k) / T(n, k), (with Bernoulli(1) = 1/2).

G.f.: 1/sqrt(1 - 4*x*(y + 1)). - Vladimir Kruchinin, Feb 15 2023

A046202 Distinct numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 5, 20, 30, 60, 7, 42, 105, 140, 8, 56, 168, 280, 9, 72, 252, 504, 630, 10, 90, 360, 840, 1260, 11, 110, 495, 1320, 2310, 2772, 132, 660, 1980, 3960, 5544, 13, 156, 858, 2860, 6435, 10296, 12012, 14, 182, 1092, 4004, 10010, 18018, 24024, 15

Offset: 1

Mohammad K. Azarian

Numbers in the order in which they appear in Leibniz's Harmonic Triangle (A003506). This sequence is a permutation of the natural numbers. - Robert G. Wilson v, Jun 12 2014

			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.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A003506, A046201, A014631.

t[n_, k_] := Denominator[n!*k!/(n + k + 1)!]; DeleteDuplicates@ Flatten@ Table[t[n - k, k], {n, 0, 14}, {k, 0, n/2}] (* Robert G. Wilson v, Jun 12 2014 *)

More terms from James Sellers, Dec 13 1999

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

A132751 Triangle T(n, k) = 2/Beta(n-k+1, k) - 1, read by rows.

Original entry on oeis.org

1, 3, 3, 5, 11, 5, 7, 23, 23, 7, 9, 39, 59, 39, 9, 11, 59, 119, 119, 59, 11, 13, 83, 209, 279, 209, 83, 13, 15, 111, 335, 559, 559, 335, 111, 15, 17, 143, 503, 1007, 1259, 1007, 503, 143, 17, 19, 179, 719, 1679, 2519, 2519, 1679, 719, 179, 19

Offset: 1

Gary W. Adamson, Aug 28 2007

			First few rows of the triangle are:
   1;
   3,   3;
   5,  11,   5;
   7,  23,  23,   7;
   9,  39,  59,  39,   9;
  11,  59, 119, 119,  59,  11;
  13,  83, 209, 279, 209,  83,  13;
  15, 111, 335, 559, 559, 335, 111, 15;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A003506, A066524.

A132751:= func< n,k | 2*Factorial(n)/(Factorial(k-1)*Factorial(n-k)) -1 >;
[A132751(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 16 2021

T[n_, k_]:= 2/Beta[n-k+1, k] - 1;
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

def A132751(n, k): return 2/beta(n-k+1, k) - 1
flatten([[A132751(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 16 2021

T(n, k) = 2*A003506(n, k) - 1, an infinite lower triangular matrix.
From G. C. Greubel, Feb 16 2021: (Start)
T(n, k) = 2/Beta(n-k+1, k) - 1.
Sum_{k=1..n} T(n, k) = n*(2^n -1) = A066524(n). (End)

A132776 A128064 (unsigned) * A007318.

Original entry on oeis.org

1, 3, 2, 5, 8, 3, 7, 18, 15, 4, 9, 32, 42, 24, 5, 11, 50, 90, 80, 35, 6, 13, 72, 165, 200, 135, 48, 7, 15, 98, 273, 420, 385, 210, 63, 8, 17, 128, 420, 784, 910, 672, 308, 80, 9, 19, 162, 612, 1344, 1890, 1764, 1092, 432, 99, 10

Offset: 0

Gary W. Adamson, Aug 29 2007

Row sums = A053220: (1, 5, 16, 44, 112, 272, ...).

A003506 = A007318 * A128064 (unsigned).

			First few rows of the triangle:
   1;
   3,  2;
   5,  8,  3;
   7, 18, 15,  4;
   9, 32, 42, 24,  5;
  11, 50, 90, 80, 35,  6;
  ...

Cf. A007318, A128064, A053220, A003506.

A128064 (unsigned) * A007318 as infinite lower triangular matrices.

A187791 Repeat n+1 times 2^A005187(n).

Original entry on oeis.org

1, 2, 2, 8, 8, 8, 16, 16, 16, 16, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 32768, 32768, 32768, 32768, 32768, 32768, 32768, 32768, 32768, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536

Offset: 0

Paul Curtz, Jan 06 2013

a(n) is the denominators of the antidiagonals of the Lorentz factor, which can be written A001790(n)/A046161(n), and its differences.
1,           1/2,     3/8,      5/16,    35/128,     63/256,...  the Lorentz gamma factor,
-1/2,       -1/8,   -1/16,    -5/128,    -7/256,   -21/1024, ...  -A098597(n)/A046161(n+1),from the Lorentz (beta) factor,
3/8,        1/16,   3/128,     3/256,    7/1024,     9/2048,...  A161200(n+2)/A046161(n+2),
-5/16,    -5/128,  -3/256,   -5/1024,   -5/2048,  -45/32768,...  A161202(n+3)/A046161(n+4),
35/128,    7/256,  7/1024,    5/2048,  35/32768,   35/65536, ...
-63/256, -21/1024, -9/2048, -45/32768, -35/65536, -63/262144, ...  .
Like 1/n and A164555(n)/A027642(n), the Lorentz factor is an autosequence of the second kind. The first column is the signed sequence.
The main diagonal is (-1)^n *A001790(n)/A061549(n).
The Lorentz factor is the differences of (0, followed by A001803(n)) / (1, followed by A046161(n)).
PiSK(n-2)=(0, 0, followed by A001803(n)) / (1, 1, followed by A046161(n)) is also an autosequence of second kind.
Remember that an autosequence of the second kind is a sequence whose inverse binomial transform is the sequence signed, with its main diagonal being the double of its first upper diagonal. - Paul Curtz, Oct 13 2013

			1,
2,   2,
8,   8,  8,
16, 16, 16, 16.

OEIS Wiki, Autosequence
Wikipedia, Lorentz Factor.

Cf. A003506.

Flatten[Table[Denominator[Binomial[2n, n]/4^n], {n, 0, 19}, {n + 1}]] (* Alonso del Arte, Jan 07 2013 *)
(* Checking with the antidiagonals *) diff = Table[ Differences[ CoefficientList[ Series[1/Sqrt[1 - x], {x, 0, 9}], x], n], {n, 0, 9}]; Table[ diff[[n-k+1,k]] // Denominator,{n,0,10},{k,1,n}] // Flatten (* Jean-François Alcover, Jan 07 2013 *)
Flatten[Table[2^IntegerExponent[(2*n)!, 2], {n, 0, 19}, {n + 1}]]; (* Jean-François Alcover, Mar 27 2013, after A005187 *)

Repeat A046161(n) n+1 times. Triangle.

New definition by M. F. Hasler

A351583 Triangle read by rows: T(n,k) = A(k,n-k), 1 <= k < n, 2 <= n, where A(m,n) is the number of distinct strings consisting of one X, 2m-1 Y's and 2n-1 Z's in which the X lies to the right of at least m Y's and at least n Z's.

Original entry on oeis.org

2, 7, 7, 15, 52, 15, 26, 192, 192, 26, 40, 510, 1086, 510, 40, 57, 1115, 4098, 4098, 1115, 57, 77, 2142, 12075, 20840, 12075, 2142, 77, 100, 3752, 30072, 79600, 79600, 30072, 3752, 100, 126, 6132, 66276, 249408, 382510, 249408, 66276, 6132, 126

Offset: 2

Christopher J. Fewster, Feb 14 2022

The general string enumeration problem of counting strings with k+k'-1 X's, m+m' Y's and n+n' Z's in which the k'th X is placed after at least m of the Y's and n of the Z's may be expressed in terms of an integral of incomplete Beta functions and evaluated in terms of Kampe de Feriet functions (see Connor & Fewster, 2022). Other special cases include A351584 and A351585.

			Triangle starts:
   2;
   7,   7;
  15,  52,   15;
  26, 192,  192,  26;
  40, 510, 1086, 510, 40;
  ...

Stephen B. Connor and Christopher J. Fewster, Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration, Brazilian Journal of Probability and Statistics 2022, Vol. 36, No. 1, 185-198; arXiv version, arXiv:2104.12216 [math.CA], 2021.

Cf. A003506 (1/Beta), A005449 (column k=1), A351584, A351585.

T:=(n,k) -> 1/(2*Beta(2*k, 2*n - 2*k)) - binomial(n, k)/(2*Beta(k, n - k)); [seq(seq(T(n,k),k=1..n-1),n=2..10)];

t[n_,k_]:=1/(2*Beta[2*k,2*n-2*k])-Binomial[n,k]/(2*Beta[k,n-k]); Table[t[n,k],{n,2,10},{k,1,n-1}]

T(n+1,1) = A(1,n) = 1/2*n*(3*n+1) = A005449(n), the n-th second pentagonal number.

T(n,k) = 1/(2*Beta(2*k, 2*n - 2*k)) - binomial(n, k)/(2*Beta(k, n - k)), where Beta(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y) is the Beta-function (see A003506). [Connor and Fewster]

A132778 Triangle read by rows, n-1 terms of (2n - 1) followed by n.

Original entry on oeis.org

1, 3, 2, 5, 5, 3, 7, 7, 7, 4, 9, 9, 9, 9, 5, 11, 11, 11, 11, 11, 6, 13, 13, 13, 13, 13, 13, 7, 15, 15, 15, 15, 15, 15, 15, 8, 17, 17, 17, 17, 17, 17, 17, 17, 9, 19, 19, 19, 19, 19, 19, 19, 19, 19, 10, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 11, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 12

Offset: 1

Gary W. Adamson, Aug 29 2007

Row sums = A001844: (1, 5, 13, 25, 41, 61, ...).

			First few rows of the triangle:
  1;
  3, 2;
  5, 5, 3;
  7, 7, 7, 4;
  9, 9, 9, 9, 5;
  ...

Cf. A001844, A003506, A007318.

A007318^(-1) * [A003506 * A000012], as infinite lower triangular matrices.

Spurious extra term after a(44) deleted and more terms from Georg Fischer, Jun 02 2023

cp's OEIS Frontend

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A241519 Denominators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A356546 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

SageMath

Formula

A046202 Distinct numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A132751 Triangle T(n, k) = 2/Beta(n-k+1, k) - 1, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A132776 A128064 (unsigned) * A007318.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A351583 Triangle read by rows: T(n,k) = A(k,n-k), 1 <= k < n, 2 <= n, where A(m,n) is the number of distinct strings consisting of one X, 2m-1 Y's and 2n-1 Z's in which the X lies to the right of at least m Y's and at least n Z's.