cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A008969 Triangle of differences of reciprocals of unity.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  1,      3;
  1,     11,         7;
  1,     50,        85,         15;
  1,    274,      1660,        575,        31;
  1,   1764,     48076,      46760,      3661,       63;
  1,  13068,   1942416,    6998824,   1217776,    22631,    127;
  1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255;
  ...

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.

Links

Crossrefs

Cf. A001236, A001237, A001238, A001240, A001241, A001242.
Columns include A000254, A000424, A001236, A001237, A001238. Right-hand columns include A000225, A001240, A001241, A001242.

Programs

  • Maple
    T:= (n,k)-> `if`(k<=n, (n-k+2)!^k *
     add((-1)^(j+1)*binomial(n-k+2, j)/ j^k, j=1..n-k+2), 0):
seq(seq(T(n,k), k=0..n), n=0..7); # Alois P. Heinz, Sep 05 2008
  • Mathematica
    T[n_, k_] := If[k <= n, (n-k+2)!^k*Sum[(-1)^(j+1)*Binomial[n-k+2, j]/j^k, {j, 1, n-k+2}], 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

A112492 Triangle from inverse scaled Pochhammer symbols.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Sep 12 2005

Keywords

Comments

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)

Examples

			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

References

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

Links

Crossrefs

Row sums give A111885.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    {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
  • PARI
    /* 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
  • PARI
    /* 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
  • SageMath
    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

Formula

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

Extensions

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

A021029 Expansion of 1/((1-x)*(1-2*x)*(1-3*x)*(1-6*x)).

Original entry on oeis.org

1, 12, 97, 672, 4333, 26964, 164809, 998184, 6017605, 36192156, 217414561, 1305276336, 7834033117, 47011340388, 282089500153, 1692601439928, 10155802087669, 60935393132460, 365614101138385
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the area of the (n+3)-gon with vertices (2^k,3^k) for 0 <= k <= n+2. - J. M. Bergot and Robert Israel, Dec 05 2020

Links

Crossrefs

Cf. A001240 (first differences).

Programs

  • Magma
    [(-1+5*2^(n+2)-5*3^(n+2)+6^(n+2))/10: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011
  • Maple
    seq(-1/10 + 2^(n+1) - (9*3^n)/2 + (18*6^n)/5,n=0..40); # Robert Israel, Dec 05 2020
  • Mathematica
    CoefficientList[Series[1/((1 - x)(1 - 2x)(1 - 3x)(1 - 6x)), {x, 0, 30}], x]  (* Harvey P. Dale, Mar 14 2011 *)

Formula

G.f.: 1/((1-x)*(1-2*x)*(1-3*x)*(1-6*x)).
a(n) = (-1+5*2^(n+2)-5*3^(n+2)+6^(n+2))/10. - Bruno Berselli, Sep 02 2011

A257894 Square array read by ascending antidiagonals where T(n,k) is the mean number of maxima in a set of n random k-dimensional real vectors (numerators).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 25, 85, 15, 1, 1, 137, 415, 575, 31, 1, 1, 49, 12019, 5845, 3661, 63, 1, 1, 363, 13489, 874853, 76111, 22631, 127, 1, 1, 761, 726301, 336581, 58067611, 952525, 137845, 255, 1, 1, 7129, 3144919, 129973303, 68165041
Offset: 1

Views

Author

Jean-François Alcover, May 12 2015

Keywords

Examples

			Array of fractions begins:
1,      1,          1,             1,                 1,                    1, ...
1,    3/2,        7/4,          15/8,             31/16,                63/32, ...
1,   11/6,      85/36,       575/216,         3661/1296,           22631/7776, ...
1,  25/12,    415/144,     5845/1728,       76111/20736,        952525/248832, ...
1, 137/60, 12019/3600, 874853/216000, 58067611/12960000, 3673451957/777600000, ...
1,  49/20, 13489/3600,  336581/72000, 68165041/12960000,   483900263/86400000, ...
...
Row 2 (numerators) is A000225 (Mersenne numbers 2^k-1),
Row 3 is A001240 (Differences of reciprocals of unity),
Row 4 is A028037,
Row 5 is A103878,
Row 6 is not in the OEIS.
Column 2 (numerators) is A001008 (Wolstenholme numbers: numerator of harmonic number),
Column 3 is A027459,
Column 4 is A027462,
Column 5 is A072913,
Column 6 is not in the OEIS.

Links

Crossrefs

Cf. A257895 (denominators).

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^(j - 1)*j^(1 - k)*Binomial[n, j], {j, 1, n}]; Table[T[n - k + 1, k] // Numerator, {n, 1, 12}, {k, 1, n}] // Flatten

Formula

T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*C(n,j).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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