A003506 -id:A003506 - cp's OEIS frontend

A046207 Numbers to the right of the central elements in the triangle of denominators in Leibniz's Harmonic Triangle.

2, 3, 12, 4, 20, 5, 60, 30, 6, 105, 42, 7, 280, 168, 56, 8, 504, 252, 72, 9, 1260, 840, 360, 90, 10, 2310, 1320, 495, 110, 11, 5544, 3960, 1980, 660, 132, 12, 10296, 6435, 2860, 858, 156, 13, 24024, 18018, 10010, 4004, 1092, 182, 14, 45045, 30030, 15015, 5460

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

A177947 A symmetrical triangle sequence based on the beta function inverse and the spotlight tile A051601 as antidiagonal: t(n,m) = 1/Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}] - (-2 Binomial[m + n, m] + Binomial[2 + m + n, 1 + m]).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 22, 13, 1, 1, 19, 45, 45, 19, 1, 1, 26, 79, 110, 79, 26, 1, 1, 34, 126, 224, 224, 126, 34, 1, 1, 43, 188, 406, 518, 406, 188, 43, 1, 1, 53, 267, 678, 1050, 1050, 678, 267, 53, 1

Offset: 0

Roger L. Bagula, May 15 2010

Beta[n+1,m+1] = Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}].
Row sums are {1, 2, 6, 18, 50, 130, 322, 770, 1794, 4098, ...}.
The triangle modulo 2 is Sierpinski:
ListDensityPlot[Table[Table[Mod[ t[n, m], 2], {m, 0, 64}], {n, 0, 64}], Frame -> False, Mesh -> False].

			{1},
{1, 1},
{1, 4, 1},
{1, 8, 8, 1},
{1, 13, 22, 13, 1},
{1, 19, 45, 45, 19, 1},
{1, 26, 79, 110, 79, 26, 1},
{1, 34, 126, 224, 224, 126, 34, 1},
{1, 43, 188, 406, 518, 406, 188, 43, 1},
{1, 53, 267, 678, 1050, 1050, 678, 267, 53, 1}

Cf. A051601, A003506.

Clear[t, n]
t[n_, m_] = 1/Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}] - (-2 Binomial[m + n, m] + Binomial[2 + m + n, 1 + m]);
a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

t(n,m) = 1/Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}] - (-2 Binomial[m + n, m] + Binomial[2 + m + n, 1 + m]);

out_n,m = antidiagonal(t(n,m)) = A003506(n,m) - A051601(n,m).

A182397 Numerators in triangle that leads to the (first) Bernoulli numbers A027641/A027642.

Original entry on oeis.org

1, 1, -3, 1, -5, 5, 1, -7, 25, -5, 1, -9, 23, -35, 49, 1, -11, 73, -27, 112, -49, 1, -13, 53, -77, 629, -91, 58, 1, -15, 145, -130, 1399, -451, 753, -58, 1, -17, 95, -135, 2699, -2301, 8573, -869, 341, 1, -19, 241

Offset: 0

Paul Curtz, Apr 27 2012

In A190339 we saw that (the second Bernoulli numbers) A164555/A027642 is an eigensequence (its inverse binomial transform is the sequence signed) of the second kind, see A192456/A191302. We consider this array preceded by 1 for the second row, by 1, -3/2, for the third one; 1 is chosen and is followed by the differences of successive rows.
Hence
                    1    1/2   1/6      0
            1    -1/2   -1/3  -1/6  -1/30
      1   -3/2    1/6    1/6  2/15   1/15
  1 -5/2   5/3      0  -1/30 -1/15 -8/105.
The second row is A051716/A051717.
The (reduced) triangle before the square array (T(n,m) in A190339) is a(n)/b(n)=
B(0)=    1 = 1                 Redbernou1li
B(1)= -1/2 = 1  -3/2
B(2)=  1/6 = 1  -5/2  5/3
B(3)=    0 = 1  -7/2 25/6  -5/3
B(4)=-1/30 = 1  -9/2 23/3 -35/6  49/30
B(5)=    0 = 1 -11/2 73/6 -27/2 112/15 -49/30.
For the main diagonal, see A165142.
Denominator b(n) will be submitted.
This transform is valuable for every eigensequence of the second kind. For instance Leibniz's 1/n (A003506).
With increasing exponents for coefficients, polynomials CB(n,x) create Redbernou1li. See the formula.
Triangle Bernou1li for A027641/A027642 with the same denominator A080326 for every column is
1
1  -3/2
1  -5/2 10/6
1  -7/2 25/6 -10/6
1  -9/2 46/6 -35/6  49/30
1 -11/2 73/6 -81/6 224/30 -49/30.
For numerator by columns,see A000012, -A144396, A100536, Q(n)=n*(2*n^2+9*n+9)/2 , new.
Triangle Checkbernou1 with the same denominator A080326 for every row is
1/1
(2    -3)/2
(6   -15  +10)/6
(6   -21  +25  -10)/6
(30 -135 +230 -175  +49)/30
(30 -165 +365 -405 +224 -49)/30;
Hence for numerator: 1, 2-3, 16-15, 31-31, 309-310, 619-619, 8171-8166.
Absolute sum: 1, 5, 31, 62, 619, 1238, 17337. Reduced division by A080326:
1, 5/2, 31/6, 31/3, 619/30, 619/15, 5779/70, = A172030(n+1)/A172031(n+1).

Cf. A028246 (Worpitzky), A085737/A085738 (Conway-Sloane), A051714/A051715 (Akiyama-Tanigawa), A192456/A191302 for other triangles that lead to the Bernoulli numbers.

CB(0,x) = 1,
CB(1,x) = 1 - 3*x/2,
CB(n,x) = (1-x)*CB(n-1,x) + B(n)*x^n , n > 1.

A273416 Numbers that occur 5 or more times in the triangle of denominators in Leibniz's Harmonic Triangle.

Original entry on oeis.org

30, 1260, 1980, 74256, 1861860, 603380175840

Offset: 1

Gionata Neri, May 22 2016

Of the first 6 terms the number 30 occurs 5 times, the others occur 6 times.

a(6) = 603380175840 is not an oblong number.

			The number 30 appears 1 time in row 5 in the central position {5*C(4,2)}, 2 times in row 6 {6*C(5,1); 6*C(5,4)} and, trivially, 2 times in row 30 {30*C(29,0); 30*C(29,29)}.
The number 1260 appears 2 times in row 10 {10*C(9,4); 10*C(9,5)}, 2 times in row 36 {36*C(35,1); 36*C(35,34)} and, trivially, 2 times in row 1260 {1260*C(1259,0); 1260*C(1259,1259)}.

Cf. A003506.

A316140 Denominator of the autosequence 2/((n+2)*(n+3)) difference table written by antidiagonals.

Original entry on oeis.org

3, 6, 6, 10, 15, 10, 15, 30, 30, 15, 21, 105, 70, 105, 21, 28, 84, 140, 140, 84, 28, 36, 126, 252, 315, 252, 126, 36, 45, 180, 420, 630, 630, 420, 180, 45, 55, 495, 660, 1155, 1386, 1155, 660, 495, 55, 66, 330

Offset: 0

Paul Curtz, Jun 25 2018

			Difference table:
   1/3,   1/6,    1/10,   1/15,  ...
  -1/6,  -1/15,  -1/30,  -2/105, ...
   1/10,  1/30,   1/70,   1/140, ...
  -1/15, -2/105, -1/140, -1/315, ... .
  ...
Table starts:
   3   6   10    15    21    28   ...
   6  15   30   105    84   126   ...
  10  30   70   140   252   420   ...
  15 105  140   315   630  1155   ...
  21  84  252   630  1386  2772   ...
  ...
As a triangle:
   3;
   6,  6;
  10, 15, 10;
  15, 30, 30, 15;
  ...

OEIS Wiki, Autosequence

Cf. A000217, A003506, A033876? (main diagonal), A059481, A109613.

tabl(nn) = {nn = 2*nn; m = matrix(nn, nn, n, k, if (n==1, 2/((k+1)*(k+2)))); for (n=2, nn, for (k=1, nn-n +1, m[n, k] = m[n-1, k+1] - m[n-1,k];);); nn = nn/2; matrix(nn, nn, n, k, denominator(m[n,k]));} \\ Michel Marcus, Jul 05 2018

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A046207 Numbers to the right of the central elements in the triangle of denominators in Leibniz's Harmonic Triangle.

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A177947 A symmetrical triangle sequence based on the beta function inverse and the spotlight tile A051601 as antidiagonal: t(n,m) = 1/Integrate[(-1 + t)^n/t^(m + n + 2), {t, 1, Infinity}] - (-2 Binomial[m + n, m] + Binomial[2 + m + n, 1 + m]).

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A182397 Numerators in triangle that leads to the (first) Bernoulli numbers A027641/A027642.

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A273416 Numbers that occur 5 or more times in the triangle of denominators in Leibniz's Harmonic Triangle.

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A316140 Denominator of the autosequence 2/((n+2)*(n+3)) difference table written by antidiagonals.

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