A091137 -id:A091137 - cp's OEIS frontend

A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.

-1, 1, 1, 1, 19, 27, 863, 1375, 33953, 57281, 3250433, 5675265, 13695779093, 24466579093, 132282840127, 240208245823, 111956703448001, 205804074290625, 151711881512390095, 281550972898020815, 86560056264289860203, 161867055619224199787, 20953816286242674495191, 39427936010479474495191

Offset: 0

Paul Curtz, Aug 05 2008

sign

This is row i=0 of an array defined as T(i,j) = (-1)^(i+j+1)*A091137(j)*a(i,j), columns j >= 0, which starts
  -1,   1,   1,    1,    19,    27,   863, ...
   1,  -3,   5,    1,    11,    11,   271, ...
  -1,   5, -23,    9,    19,    11,   191, ...
   1,  -7,  53,  -55,   251,    27,   271, ...
  -1,   9, -95,  161, -1901,   475,   863, ...
   1, -11, 149, -351,  6731, -4277, 19087, ...
  ...
The first two rows are related via T(0,j) = A027760(j)*T(0,j-1) - T(1,j).

P. Curtz, Integration .., note 12, C.C.S.A., Arcueil, 1969.

Cf. A141047, A140811, A140825.

A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end proc:
A048994 := proc(n, k) combinat[stirling1](n, k) ; end proc:
a := proc(i,j) add(A048994(j,k)*x^k,k=0..j) ; int(%,x=i..i+1) ; %/j! ; end proc:
A141417 := proc(n) (-1)^(n+1)*A091137(n)*a(0,n) ; end proc:
seq(A141417(n),n=0..40) ; # R. J. Mathar, Nov 17 2010

(* a7 = A091137 *) a7[n_] := a7[n] = Times @@ Select[ Divisors[n]+1, PrimeQ]*a7[n-1]; a7[0]=1; a[n_] := (-1)^(n+1) * a7[n] * Integrate[ (-1)^n*Pochhammer[-x, n], {x, 0, 1}]/n!; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Aug 10 2012 *)

a(n):=if n=0 then -1 else num(n*(n+1)*sum(((-1)^(n-k)*stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1)),k,1,n)); /* Vladimir Kruchinin, Dec 12 2016 */

a(i,j) = a(i-1,j) + a(i-1,j-1), see reference page 33.
(q+1-j)*Sum_{j=0..q} a(i,j)*(-1)^(q-j) = binomial(i,q), see reference page 35.
a(n) = numerator(n*(n+1)*Sum_{k=1..n} ((-1)^(n-k)*Stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1))), n>0, a(0)=-1. - Vladimir Kruchinin, Dec 12 2016

Erroneous formula linking A091137 and A002196 removed, and more terms and program added by R. J. Mathar, Nov 17 2010

A141047 Numerators of A091137(n)T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x(x-1)(x-2) .. *(x-j+1) dx.

Original entry on oeis.org

1, 3, 23, 55, 1901, 4277, 198721, 434241, 14097247, 30277247, 2132509567, 4527766399, 13064406523627, 27511554976875, 173233498598849, 362555126427073, 192996103681340479, 401972381695456831, 333374427829017307697, 691668239157222107697, 236387355420350878139797

Offset: 0

Paul Curtz, Jul 31 2008

nonn

Numerators of the main diagonal of the array A091137(j)*T(i,j)/j! where T(i,j)=Integral (x= i.. i+1) x*(x-1)*(x-2)* .. *(x-j+1) dx.
The reduced fractions of the array T(i,j) are shown in A140825, which also describes how the integrand is a generating function of Stirling numbers.
The sequence A027760 plays a role i) in relating to A091137 as described there and
ii) in a(n+1)-A027760(n+1)*a(n)= A002657(n+1), numerators of the diagonal T(n,n+1).

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note 12, Centre de Calcul Scientifique de l' Armement, Arcueil (1969), p. 36.

Cf. A140811, A140825, A141045.

T := proc(i,j) local var,k ; var := x ; for k from 1 to j-1 do var := var*(x-k) ; od: int(var,x=i..i+1) ; simplify(A091137(j)*%/j!) ; numer(%) ; end:
A141047 := proc(n) T(n,n) ; end: for n from 0 to 20 do printf("%a,",A141047(n) ) ; od: # R. J. Mathar, Feb 23 2009

b[n_] := b[n] = (* A091137 *) If[n==0, 1, Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*b[n-1]]; T[i_, j_] := Integrate[Product[x-k, {k, 0, j-1}], {x, i, i+1}]; a[n_] := b[n]*T[n, n]/n!; Table[a[n] // Numerator, {n, 0, 20}] (* Jean-François Alcover, Jan 10 2016 *)

a(n) = numerator( A091137(n)*T(n,n)/n!) where T(n,n) = sum_{k=0..n} A048994(n,k)*((n+1)^(k+1)-n^(k+1))/(k+1).

Edited and extended by R. J. Mathar, Feb 23 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 *)

A145178 Numerators of coefficients of series expansion of 1/(Bernoulli trial entropy), scaled to denominators A091137.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 36, 80, 90, 45, 48, 112, 150, 120, 45, 1440, 3444, 5068, 5040, 3150, 945, 2160, 5232, 8148, 9184, 7350, 3780, 945, 50400, 122832, 198360, 242200, 224700, 151200, 66150, 14175, 80640, 196992, 325896, 420160, 429800, 341040, 198450

Offset: 1

Tilman Neumann, Oct 03 2008

These are the numerators A145176 scaled to denominators A091137.

In other words: A145178(n,k)/A091137(n) = A145176(n,k)/A145177(n,k)

Cf. A091137, A145176, A145177

Corrected description of the series. - Tilman Neumann, Oct 04 2008

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

A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u(u+1)...*(u+n-1) du.

Original entry on oeis.org

2, 0, 4, 8, 232, 448, 18224, 35424, 1036064, 2025472, 130960832, 257072000, 689908475264, 1358275350528, 8031885897472, 15847920983552, 7981032500085248, 15774370258485248, 12448755354530366464

Offset: 0

Paul Curtz, Sep 20 2011

nonn

Numerators of the second row of an array based on Adams numerical integration. Take q!*s(m,q) = Integral_{-m-1..1} u*(u+1)*...*(u+q-1) du. a(n) is in the second row (case m=0) numerators of s(m,q) in the comments.
The unreduced array s(m,q), (m=-1,0,1,...,   columns q=0,1,2,...) is
1,   1/2,   5/12,   9/24,   251/720, 475/1440,  = A002657(n)/A091137(n),
2,     0,   4/12,   8/24,   232/720, 448/1440,  = a(n)/A091137(n),
3,  -3/2,   9/12,   9/24,   243/720, 459/1440,
4,  -8/2,  32/12,      0,   224/720, 448/1440,
5, -15/2,  85/12, -55/24,   475/720, 475/1440,
6, -24/2, 180/12, -216/24, 2376/720,        0.
Column numerators: A000027, -A067998(n), A152064(n), A157371(n), A165281(n).
Page 56 of the reference.
(*) 2/2 = 1,
    2/2 + 0 = 1,
    2/3 + 0 + 1/3 = 1,
    2/4 + 0 + 1/6 + 1/3 = 1. Reduced.

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

A195287 := proc(n)
        mul(u+i,i=0..n-1) ;
        int(%,u=-1..1) ;
        %/n!*A091137(n) ;
end proc:
seq(A195287(n),n=0..20) ; # R. J. Mathar, Oct 02 2011

(* a7 = A091137 *) a7[n_] := a7[n] = Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*a7[n-1]; a7[0]=1; a[n_] := a7[n]/n!*Integrate[ Pochhammer[u, n], {u, -1, 1}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 13 2012 *)

b(n) = a(n)/A091137(n).
b(0)/2 = 1,
b(0)/2 + b(1) = 1,
b(0)/3 + b(1)/2 + b(2) = 1,
b(0)/4 + b(1)/3 + b(2)/2 + b(3) = 1.
First vertical denominators: A028310(n) + 1. See A104661.
Values in (*).

A141421 First bisection of A091137.

Original entry on oeis.org

1, 12, 720, 60480, 3628800, 479001600, 2615348736000, 31384184832000, 32011868528640000, 51090942171709440000, 33720021833328230400000, 9306726025998591590400000, 50814724101952310083584000000

Offset: 0

Paul Curtz, Aug 05 2008

nonn

The second bisection is obtained by doubling the terms, 2*a(n).

A091137 := proc(n) local a,p ; a := 1; p := 2: while floor(n/(p-1)) > 0 do a := a*p^(floor(n/(p-1))) ; p := nextprime(p) ; od: RETURN(a) ; end: A141421 := proc(n) A091137(2*n) ; end: for n from 0 to 20 do printf("%d,",A141421(n)) ; od: # R. J. Mathar, Aug 09 2008

a(n)=A091137(2n).

Edited and extended by R. J. Mathar, Aug 09 2008

A165636 a(n) = A091137(n)/2^n.

Original entry on oeis.org

1, 1, 3, 3, 45, 45, 945, 945, 14175, 14175, 467775, 467775, 638512875, 638512875, 1915538625, 1915538625, 488462349375, 488462349375, 194896477400625, 194896477400625, 32157918771103125, 32157918771103125, 2218896395206115625, 2218896395206115625, 3028793579456347828125, 3028793579456347828125, 9086380738369043484375

Offset: 0

Paul Curtz, Sep 23 2009

nonn

Cf. A171080, A001897, A036278.

A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; end if; a := a*p^floor(n/(p-1)) ; end do: a ; end proc:
A165636 := proc(n) A091137(n)/2^n ; end proc: # R. J. Mathar, Jul 07 2011

a(n)=my(p=primes(primepi(n+1)));prod(i=1,#p,p[i]^(n\(p[i]-1)))>>n \\ Charles R Greathouse IV, Jul 07 2011

a(n+1) = a(n)* A141459(n+1).

A140783 Digit sum of A091137(n).

Original entry on oeis.org

1, 2, 3, 6, 9, 9, 18, 18, 27, 27, 27, 27, 45, 45, 45, 63, 54, 54, 63, 72, 54, 81, 99, 90, 81, 90, 99, 99, 99, 126, 117, 135, 189, 198, 171, 180, 171, 180, 171, 180, 189, 162, 189, 207, 180, 198, 207, 243, 261, 243, 297, 270, 243, 261, 279, 288, 324, 342, 315, 315

Offset: 0

Paul Curtz, Jul 13 2008

a(n)= A007953(A091137(n)). - R. J. Mathar, Jul 29 2008

Edited and extended by R. J. Mathar, Jul 29 2008

A165641 A091137(n) / A001316(n) .

Original entry on oeis.org

1, 1, 6, 6, 360, 360, 15120, 15120, 1814400, 1814400, 119750400, 119750400, 653837184000, 653837184000, 3923023104000, 3923023104000, 16005934264320000, 16005934264320000, 12772735542927360000, 12772735542927360000, 8430005458332057600000, 8430005458332057600000

Offset: 0

Paul Curtz, Sep 23 2009

nonn

Cf. A165636, A036281, A002445.

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

cp's OEIS Frontend

A141417 (-1)^(n+1)*A091137(n)*a(0,n), where a(i,j) = Integral_{x=i..i+1} x*(x-1)*(x-2)*...*(x-j+1)/j! dx.

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A141047 Numerators of A091137(n)*T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x*(x-1)*(x-2)* .. *(x-j+1) dx.

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

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A145178 Numerators of coefficients of series expansion of 1/(Bernoulli trial entropy), scaled to denominators A091137.

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

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A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u*(u+1)*...*(u+n-1) du.

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A141421 First bisection of A091137.

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A165636 a(n) = A091137(n)/2^n.

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A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.

A141047 Numerators of A091137(n)T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x(x-1)(x-2) .. *(x-j+1) dx.

A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u(u+1)...*(u+n-1) du.