A147998 -id:A147998 - cp's OEIS frontend

A165795 Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.

1, -1, 1, -1, 0, 1, -1, -3, 3, 1, -1, -8, 0, 8, 1, -1, -15, -5, 5, 15, 1, -1, -24, -3, 0, 3, 24, 1, -1, -35, -21, -7, 7, 21, 35, 1, -1, -48, -2, -16, 0, 16, 2, 48, 1, -1, -63, -45, -1, -9, 9, 1, 45, 63, 1, -1, -80, -15, -40, -5, 0, 5, 40, 15, 80, 1, -1, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, 1

Offset: 0

Paul Curtz, Sep 27 2009

A row of A(0,k)= 1 is added on top of the array shown in A172157, which is then read upwards by antidiagonals.

One may also interpret this as appending a 1 to each row of A173651 or adding a column of -1's and a diagonal of +1's to A165507.

			The array, A(n, k), of numerators starts in row n=0 with columns m>=0 as:
  .1...1...1...1...1...1...1...1...1...1...1.
  -1...0...3...8..15..24..35..48..63..80..99. A005563, A147998
  -1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262
  -1..-8..-5...0...7..16...1..40..55...8..91. A061039
Antidiagonal triangle, T(n, k), begins as:
   1;
  -1,   1;
  -1,   0,   1;
  -1,  -3,   3,  1;
  -1,  -8,   0,  8,  1;
  -1, -15,  -5,  5, 15,  1;
  -1, -24,  -3,  0,  3, 24,  1;
  -1, -35, -21, -7,  7, 21, 35, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A158388, A165507, A172157, A173651.

Cf. A005408, A005563, A061037, A061039, A070262, A147998.

T[n_, k_]:= If[k==n, 1, If[k==0, -1, Numerator[1/(n-k)^2 - 1/k^2]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 10 2022 *)

def A165795(n,k):
    if (k==n): return 1
    elif (k==0): return -1
    else: return numerator(1/(n-k)^2 -1/k^2)
flatten([[A165795(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 10 2022

A(n, k) = numerator(1/n^2 - 1/k^2) with A(0,k) = 1 and A(n,0) = -1 (array).
A(n, 0) = -A158388(n).
A(n, k) = A172157(n,k), n>=1.
From G. C. Greubel, Mar 10 2022: (Start)
T(n, k) = numerator(1/(n-k)^2 -1/k^2), with T(n,n) = 1, T(n,0) = -1 (triangle).
A(n, n) = T(2*n, n) = 0^n.
Sum_{k=0..n} T(n, k) = 0^n.
T(n, n-k) = -T(n,k).
T(2*n+1, n) = -A005408(n). (End)

A157371 a(n) = (n+1)(9-9n+5*n^2-n^3).

Original entry on oeis.org

9, 8, 9, 0, -55, -216, -567, -1216, -2295, -3960, -6391, -9792, -14391, -20440, -28215, -38016, -50167, -65016, -82935, -104320, -129591, -159192, -193591, -233280, -278775, -330616, -389367, -455616, -529975, -613080, -705591, -808192, -921591, -1046520, -1183735, -1334016, -1498167

Offset: 0

Paul Curtz, Feb 28 2009

This is the fourth in a family of sequences that appear in columns on pages 36 and 56 of the reference: (i) sequence n+1, A000029, (ii) sequence (n+1)*(1-n), A147998 and (iii) (n+1)*(5-5*n+2*n^2), A152064.

First differences along columns shown on page 56 of the reference are columns of what is shown on page 36 of the reference. Example: the third column of page 56, A152064, has first differences which constitute the third column p page 36, A140811.

Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[(n+1)*(9-9*n+5*n^2-n^3): n in [0..40] ]; // Vincenzo Librandi, Jul 14 2011

LinearRecurrence[{5,-10,10,-5,1},{9,8,9,0,-55},40] (* or *) Table[(n+1)(9-9n+5n^2-n^3),{n,0,40}] (* or *) CoefficientList[ Series[ (55x^3- 59x^2+ 37x-9)/ (x-1)^5,{x,0,40}],x] (* Harvey P. Dale, Jul 13 2011 *)

a(n)=(n+1)*(9-9*n+5*n^2-n^3) \\ Charles R Greathouse IV, Oct 16 2015

First differences: a(n+1)-a(n) = -A141530(n).
Fourth differences: a(n+4)-4*a(n+3)+6*a(n+2)-4*a(n+1)+a(n) = -24 = -A010863(n).
From Harvey P. Dale, Jul 13 2011: (Start)
a(0)=9, a(1)=8, a(2)=9, a(3)=0, a(4)=-55, a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
G.f.: (9-37*x+59*x^2-55*x^3)/(1-x)^5. (End)
E.g.f.: (9 - x + x^2 - 2*x^3 - x^4)*exp(x). - G. C. Greubel, Feb 02 2018

Edited, extended by R. J. Mathar, Sep 25 2009

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 1, 2, 12, 24, 1, 2, 12, 24, 720, 1, 2, 12, 24, 720, 1440, 1, 2, 12, 24, 720, 1440, 60480, 1, 2, 12, 24, 720, 1440, 60480, 120960, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 1, 2, 12

Offset: 1

Paul Curtz, Sep 14 2009

From a study of modified initialization formulas in Adams-Bashforth (1855-1883) multisteps method for numerical integration. On p.36, a(i,j) comes from (j!)*a(i,j) = Integral_{u=i,..,i+1} u*(u-1)*...*(u-j+1) du; see p.32.
Then, with i vertical, j horizontal, with unreduced fractions, partial array is:
0) 1,  1/2,  -1/12,   1/24,  -19/720,   27/1440, ... =  1/log(2)
1) 1,  3/2,   5/12,  -1/24,   11/720,  -11/1440, ... =  2/log(2)
2) 1,  5/2,  23/12,   9/24,  -19/720,   11/1440, ... =  4/log(2)
3) 1,  7/2,  53/12,  55/24,  251/720,  -27/1440, ... =  8/log(2)
4) 1,  9/2,  95/12, 161/24, 1901/720,  475/1440, ... = 16/log(2)
5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... = 32/log(2)
... [improved by Paul Curtz, Jul 13 2019]
First line: the reduced terms are A002206/A002207, logarithmic or Gregory numbers G(n). The difference between the second line and the first one is 0 together A002206/A002207. This is valid for the next lines. - Paul Curtz, Jul 13 2019
See A141417, A140825, A157982, horizontal numerators: A141047, vertical numerators: A000012, A005408, A140811, A141530, A157411. On p.56, coefficients are s(i,q) = (1/q!)* Integral_{u=-i-1,..,1} u*(u+1)*...*(u+q-1) du.
Unreduced fractions array is:
-1) 1,   1/2,  5/12,   9/24, 251/720, 475/1440, ... = A002657/A091137
0)  2,   0/2,  4/12,   8/24, 232/720, 448/1440, ... = A195287/A091137
1)  3,  -3/2,  9/12,   9/24, 243/720, 459/1440, ...
2)  4,  -8/2, 32/12,   0/24, 224/720, 448/1440, ...
3)  5, -15/2, 85/12, -55/24, 475/720, 475/1440, ...
...
(on p.56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281.
From Paul Curtz, Jul 14 2019: (Start)
Difference table from the second line and the first one difference:
1,                -1/2,      -1/12,    -1/24,    -19/720, -27/1440, ...
-3/2,             5/12,       1/24,    11/720,    11/1440, ...
23/12,           -9/24,    -19/720,  -11/1440, ...
-55/24,        251/720,    27/1440, ...
1901/720,    -475/1440,
-4277/1440, ...
...
Compare the lines to those of the first array.
The verticals are the signed diagonals of the first array. (End)

			1;
1,2;
1,2,12;
1,2,12,24;
1,2,12,24,720;

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.

Cf. A090624, A091137.
Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.
Cf. A002206, A002207.

(* a = A091137 *) a[n_] := a[n] = Product[d, {d, Select[Divisors[n]+1, PrimeQ]}]*a[n-1]; a[0]=1; Table[Table[a[k-1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 18 2014 *)

A174727 a(n) = A091137(n+1)/(n+1).

Original entry on oeis.org

2, 6, 8, 180, 288, 10080, 17280, 453600, 806400, 47900160, 87091200, 217945728000, 402361344000, 2241727488000, 4184557977600, 2000741783040000, 3766102179840000, 2838385676206080000, 5377993912811520000, 1686001091666411520000, 3211430650793164800000, 423033001181754163200000

Offset: 0

Paul Curtz, Mar 28 2010

nonn

Previous name: Inverse Akiyama-Tanigawa algorithm. From a column instead of a row. Bernoulli case A164555/A027642. We start from column 1, 1/2, 1/3, 1/4, 1/5 = A000012/A000027. First row: 1) (unreduced) 1, 1/2, 5/12, 9/24, 251/720 = A002657/A091137 (Cauchy from Bernoulli) (*); 2) (reduced) 1, 1/2, 5/12, 3/8, 251/720 = A002208/A002209 (Stirling and Bernoulli). Unreduced second row: 1/2, 1/6, 1/8, 19/180, 27/288, 863/10080 = A141417(n+1)/a(n).

(*) Reference page 56 (first row) and page 36 (upper main diagonal). From J. C. Adams (and Bashforth) numerical integration. See A165313 and A147998. See A002206 logarithm numbers (Gregory).

P. Curtz, Intégration numérique des systèmes différentiels .. . Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.

Cf. A002689, A091137.

A091137[n_] := A091137[n] = Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]*A091137[n-1]; A091137[0] = 1; a[n_] := A091137[n+1]/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover_, Aug 14 2012 *)

f(n) = my(r =1); forprime(p=2, n+1, r*=p^(n\(p-1))); r; \\ A091137
a(n) = f(n+1)/(n+1); \\ Michel Marcus, Jun 30 2019

a(n) = A091137(n+1)/(n+1).

Extended up to a(18) by Jean-François Alcover, Aug 14 2012

New name and more terms from Michel Marcus, Jun 30 2019

cp's OEIS Frontend

A165795 Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A157371 a(n) = (n+1)*(9-9*n+5*n^2-n^3).

Views

Author

Keywords

Comments

References

Links

Programs

Magma

Mathematica

PARI

Formula

Extensions

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A174727 a(n) = A091137(n+1)/(n+1).

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A157371 a(n) = (n+1)(9-9n+5*n^2-n^3).