A001242 -id:A001242 - cp's OEIS frontend

A112492 Triangle from inverse scaled Pochhammer symbols.

1, 1, 1, 1, 3, 1, 1, 7, 11, 1, 1, 15, 85, 50, 1, 1, 31, 575, 1660, 274, 1, 1, 63, 3661, 46760, 48076, 1764, 1, 1, 127, 22631, 1217776, 6998824, 1942416, 13068, 1, 1, 255, 137845, 30480800, 929081776, 1744835904, 104587344, 109584, 1, 1, 511, 833375, 747497920, 117550462624, 1413470290176, 673781602752, 7245893376, 1026576, 1

Offset: 0

Wolfdieter Lang, Sep 12 2005

This expansion is based on the partial fraction identity: 1/Product_{j=1..m}(x+j) = (1 + Sum_{j=1..m} (-1)^j*binomial(m,j) * x/(x+j))/m!, e.g., p. 37 of the Jordan reference.
Another version of this triangle (without a column of 1's) is A008969.
The column sequences are, for m=1..10: A000012 (powers of 1), A000225, A001240, A001241, A001242, A111886-A111888.
From Gottfried Helms, Dec 11 2001: (Start)
The triangle occurs as U-factor in the LDU-decomposition of the matrix M defined by m(r,c) = 1/(1+r)^c (r, c beginning at 0).
Then
  a(r,c) = m(r,c) * (1+r)!^(c-r).
An explicit expansion based on this can be made by defining a "recursive harmonic number" (rhn). (This representation is just a heuristic pattern-interpretation, no analytic proof yet available).
Consider
  h(k,0)=1      for k>0      as rhn of order zero(0).
Then consider
  h(1,1)=1*h(1,0)
  h(2,1)=1*h(1,0) + 1/2*h(2,0)
  h(3,1)=1*h(1,0) + 1/2*h(2,0) + 1/3*h(3,0) = h(2,1)+1/3*h(3,0)
  ...
and recursively
  h(1,r)=1*h(1,r-1)
  h(2,r)=1*h(1,r-1) + 1/2*h(2,r-1)
  h(3,r)=1*h(1,r-1) + 1/2*h(2,r-1) + 1/3*h(3,r-1) = h(2,r)+1/3*h(3,r-1)
  ...
  h(k,r)=h(k-1,r)+1/k*h(k,r-1)
then the upper triangular triangle A:=a(r,c) for c-r>0
a(r,c) = h(r,c-r) *(1+r)!^(c-r).
(End)

			Triangle begins:
  1;
  1,   1;
  1,   3,     1;
  1,   7,    11,       1;
  1,  15,    85,      50,       1;
  1,  31,   575,    1660,     274,       1;
  1,  63,  3661,   46760,   48076,    1764,     1;
  1, 127, 22631, 1217776, 6998824, 1942416, 13068,    1; ...
The g.f.s for the rows are illustrated by:
Sum_{n>=0} (n+1)^(n-1)*exp((n+1)*x)*(-x)^n/n! = 1;
Sum_{n>=0} (n+1)^(n-2)*exp((n+1)*x)*(-x)^n/n! = 1 + 1*x/2!;
Sum_{n>=0} (n+1)^(n-3)*exp((n+1)*x)*(-x)^n/n! = 1 + 3*x/2!^2 + 1*x^2/3!;
Sum_{n>=0} (n+1)^(n-4)*exp((n+1)*x)*(-x)^n/n! = 1 + 7*x/2!^3 + 11*x^2/3!^2 + 1*x^3/4!;
Sum_{n>=0} (n+1)^(n-5)*exp((n+1)*x)*(-x)^n/n! = 1 + 15*x/2!^4 + 85*x^2/3!^3 + 50*x^3/4!^2 + 1*x^4/5!; ...
which are derived from a LambertW() identity. - _Paul D. Hanna_, Oct 20 2012

Charles Jordan, Calculus of Finite Differences, Chelsea, 1965.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Lang, First 10 rows.
L. M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:0006106 [math.CO], 2000.

Row sums give A111885.

function T(n,k) // T = A112492
  if k eq 0 or k eq n then return 1;
  else return (k+1)^(n-k)*T(n-1,k-1) + Factorial(k)*T(n-1,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 24 2023

T[, 0]=1; T[n, m_]:= -m!^(n-m+1)*Sum[(-1)^j*Binomial[m, j]/j^(n-m+ 1), {j,m}]; Table[T[n, m], {n,10}, {m,0,n}]//Flatten (* Jean-François Alcover, Jul 09 2013, from 2nd formula *)

{h(n,recurse=1) = if(recurse == 0, return(1)); ;
return( sum(k=0,n, h(k,recurse-1) / (1+k) )); }
a(r,c) = h(r-1,c-r) * r!^(c-r) \\ Gottfried Helms, Dec 11 2001

/* From g.f. for column k: */
T(n,k) = (k+1)!^(n-k+1)*polcoeff(prod(j=0,k,1/(j+1-x +x*O(x^(n-k)))),n-k)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Oct 20 2012

/* From g.f. for row n: */
T(n,k) = (k+1)!^(n-k+1)*polcoeff(sum(j=0,k,(j+1)^(j-n-1)*exp((j+1)*x +x*O(x^k))*(-x)^j/j!),k)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Oct 20 2012

def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 24 2023

G.f. for column m>=1: (x^m)/product(1-m!*x/j, j=1..m).

T(n, m) = -(m!^(n-m+1))*Sum_{j=1..m} (-1)^j*binomial(m, j)/j^(n-m+1), m>=1. T(n, m)=0 if n+1

G.f. of column k: x^k/Product_{j=0..k} (j+1 - x) = Sum_{n>=k} T(n,k)*x^k/(k+1)!^(n-k+1). - Paul D. Hanna, Oct 20 2012

T(n,k) = (k+1)!^(n-k+1) * [x^n] x^k / Product_{j=0..k} (j+1 - x). - Paul D. Hanna, Oct 20 2012

G.f. of row n: Sum_{j>=0} (j+1)^(j-n-1) * exp((j+1)*x) * (-x)^j/j! = Sum_{k>=0} T(n,k)*x^k/(k+1)!^(n-k+1). - Paul D. Hanna, Oct 20 2012

T(n,k) = (k+1)!^(n-k+1) * [x^k] Sum_{j>=0} (j+1)^(j-n-1) * exp((j+1)*x) * (-x)^j/j!. - Paul D. Hanna, Oct 20 2012

T(n,0) = T(n,n) = 1 and T(n,k) = (k+1)^(n-k)*T(n-1,k-1)+(k!)*T(n-1,k) for 0Werner Schulte, Dec 14 2016

Terms a(48) onward added by G. C. Greubel, Nov 12 2017

A103878 Expansion of x / ((1-12x)(1-15x)(1-20x)(1-30x)(1-60x)).

Original entry on oeis.org

0, 1, 137, 12019, 874853, 58067611, 3673451957, 226576032859, 13790081534933, 833490615528571, 50196582942983477, 3017503478665411099, 181223386494229073813, 10878637590184410995131, 652876078076017863582197, 39177315318319288337595739

Offset: 0

Ralf Stephan, Feb 20 2005

Colin Barker, Table of n, a(n) for n = 0..562
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Index entries for linear recurrences with constant coefficients, signature (137,-6750,153000,-1620000,6480000).

Cf. A001242.

CoefficientList[Series[x/((1-12x)(1-15x)(1-20x)(1-30x)(1-60x)),{x,0,20}],x] (* or *) LinearRecurrence[ {137,-6750,153000,-1620000,6480000},{0,1,137,12019,874853},20] (* Harvey P. Dale, Aug 31 2024 *)

concat(0, Vec(x/(-6480000*x^5+1620000*x^4-153000*x^3+6750*x^2-137*x+1) + O(x^100))) \\ Colin Barker, Apr 26 2015

a(n) = (1/12) (12^n - 4*15^n + 6*20^n - 4*30^n + 60^n).

A111886 Sixth column of triangle A112492 (inverse scaled Pochhammer symbols).

Original entry on oeis.org

1, 1764, 1942416, 1744835904, 1413470290176, 1083688832185344, 806595068762689536, 590914962115587293184, 429295503918929370218496, 310518802877016005311463424, 224098118280955193084850733056

Offset: 0

Wolfdieter Lang, Sep 12 2005

Also continuation of family of differences of reciprocals of unity. See A001242 and triangle A008969.

G. C. Greubel, Table of n, a(n) for n = 0..250
Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Also right-hand column 5 in triangle A008969.

A111886:= func< n | (-1)*Factorial(6)^n*(&+[(-1)^j*Binomial(6,j)/j^n : j in [1..6]]) >;
[A111886(n): n in [0..30]]; // G. C. Greubel, Jul 24 2023

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1,k-1] +k!*T[n-1,k]]; (* T = A112492 *)
Table[T[n+5,5], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

a(n) = -((6!)^n)*sum(j=1, 6, (-1)^j*binomial(6, j)/j^n); \\ Michel Marcus, Apr 28 2020

@CachedFunction
def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
def A111886(n): return T(n+5,5)
[A111886(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

G.f.: 1/Product_{j=1..6} (1-6!*x/j).
a(n) = -((6!)^n)*Sum_{j=1..6} (-1)^j*binomial(6, j)/j^n, n >= 0.
a(n) = A112492(n+5, 6), n>=0.

A111887 Seventh column of triangle A112492 (inverse scaled Pochhammer symbols).

Original entry on oeis.org

1, 13068, 104587344, 673781602752, 3878864920694016, 21006340945438768128, 110019668725577574273024, 565858042127972959667208192, 2882220940619488483325345857536, 14605752814655604919042956624396288

Offset: 0

Wolfdieter Lang, Sep 12 2005

Also continuation of family of differences of reciprocals of unity. See A001242, A111886 and triangle A008969.

G. C. Greubel, Table of n, a(n) for n = 0..250
Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Also right-hand column 6 in triangle A008969.

Cf. A001242, A111886, A112492.

A111887:= func< n | (-1)*Factorial(7)^n*(&+[(-1)^j*Binomial(7,j)/j^n : j in [1..7]]) >;
[A111887(n): n in [0..30]]; // G. C. Greubel, Jul 24 2023

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1,k-1] + k!*T[n-1,k]]; (* T = A112492 *)
Table[T[n+6,6], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

a(n) = -((7!)^n)*sum(j=1, 7, ((-1)^j)*binomial(7, j)/j^n); \\ Michel Marcus, Apr 28 2020

@CachedFunction
def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
def A111887(n): return T(n+6,6)
[A111887(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

G.f.: 1/Product_{j=1..7} 1-7!*x/j.
a(n) = -((7!)^n) * Sum_{j=1..7} (-1)^j*binomial(7, j)/j^n, n>=0.
a(n) = A112492(n+6, 7), n>=0.

A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).

Original entry on oeis.org

1, 109584, 7245893376, 381495483224064, 17810567950611972096, 778101042571221893382144, 32762625292956765972873609216, 1351813956241264848815287984717824

Offset: 0

Wolfdieter Lang, Sep 12 2005

Also continuation of family of Differences of reciprocals of unity. See A001242, A111887 and triangle A008969.

Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Also right-hand column 7 in triangle A008969.

Cf. A001242, A111887, A112492.

A111888:= func< n | (-1)*Factorial(8)^n*(&+[(-1)^j*Binomial(8,j)/j^n : j in [1..8]]) >;
[A111888(n): n in [0..30]]; // G. C. Greubel, Jul 24 2023

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1,k-1] +k!*T[n-1,k]]; (* T = A112492 *)
Table[T[n+7,7], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

a(n) = -((8!)^n)*sum(j=1, 8, ((-1)^j)*binomial(8, j)/j^n); \\ Michel Marcus, Apr 28 2020

@CachedFunction
def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
def A111888(n): return T(n+7,7)
[A111888(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

G.f.: 1/Product_{j=1..8} 1-8!*x/j.
a(n) = -((8!)^n) * Sum_{j=1..8} (-1)^j*binomial(8, j)/j^n, n>=0.
a(n) = A112492(n+7, 8), n>=0.

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A008969 Triangle of differences of reciprocals of unity.

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A112492 Triangle from inverse scaled Pochhammer symbols.

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A103878 Expansion of x / ((1-12x)(1-15x)(1-20x)(1-30x)(1-60x)).

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A111886 Sixth column of triangle A112492 (inverse scaled Pochhammer symbols).

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A111887 Seventh column of triangle A112492 (inverse scaled Pochhammer symbols).

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A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).

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