A033876 -id:A033876 - cp's OEIS frontend

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

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

A327809 Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)(2n+1)binomial(2n, n)Sum_{i=0..n} x^ibinomial(n, i)/(n+i+1).

Original entry on oeis.org

-1, 3, 2, -10, -15, -6, 35, 84, 70, 20, -126, -420, -540, -315, -70, 462, 1980, 3465, 3080, 1386, 252, -1716, -9009, -20020, -24024, -16380, -6006, -924, 6435, 40040, 108108, 163800, 150150, 83160, 25740, 3432, -24310, -175032, -556920, -1021020, -1178100, -875160, -408408, -109395, -12870

Offset: 0

Michel Marcus, Sep 26 2019

			Triangle begins:
     -1;
      3,     2;
    -10,   -15,     -6;
     35,    84,     70,     20;
   -126,  -420,   -540,   -315,   -70;
    462,  1980,   3465,   3080,   1386,   252;
  -1716, -9009, -20020, -24024, -16380, -6006, -924;
  ...

Karl Dilcher, Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1)=1, arXiv:1909.11222 [math.NT], 2019. See Pn(x) Table 1 p. 2.

Cf. A046899 (Q(x) polynomials, up to sign).

Cf. A001700 (1st column, up to sign), A033876 (right diagonal, up to sign).

pol(n) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*sum(i=0, n, x^i*binomial(n, i)/(n+i+1));
row(n) = Vecrev(pol(n));
tabl(nn) = for (n=0, nn, print(row(n)));

cp's OEIS Frontend

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

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A327809 Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)(2n+1)binomial(2n, n)Sum_{i=0..n} x^ibinomial(n, i)/(n+i+1).

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cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A327809 Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*Sum_{i=0..n} x^i*binomial(n, i)/(n+i+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A327809 Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)(2n+1)binomial(2n, n)Sum_{i=0..n} x^ibinomial(n, i)/(n+i+1).