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.

A210554 Triangle of coefficients of polynomials v(n,x) jointly generated with A208341; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 5, 4, 4, 9, 12, 8, 5, 14, 25, 28, 16, 6, 20, 44, 66, 64, 32, 7, 27, 70, 129, 168, 144, 64, 8, 35, 104, 225, 360, 416, 320, 128, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512
Offset: 1

Views

Author

Clark Kimberling, Mar 22 2012

Keywords

Comments

For a discussion and guide to related arrays, see A208510.
Also the number of multisets of size k that fit within some normal multiset of size n. A multiset is normal if it spans an initial interval of positive integers. - Andrew Howroyd, Sep 18 2018

Examples

			Triangle begins:
  1;
  2,  2;
  3,  5,   4;
  4,  9,  12,   8;
  5, 14,  25,  28,  16;
  6, 20,  44,  66,  64,  32;
  7, 27,  70, 129, 168, 144, 64;
  ...
First three polynomials v(n,x): 1, 2 + 2x , 3 + 5x + 4x^2.
The T(3, 1) = 3 multisets: (1), (2), (3).
The T(3, 2) = 5 multisets: (11), (12), (13), (22), (23).
The T(3, 3) = 4 multisets: (111), (112), (122), (123).

Links

Crossrefs

Row sums are A027941.
Cf. A160232, A208341, A208510, A303974.

Programs

  • Maple
    T := (n,k) -> simplify((n + 1 - k)*hypergeom([1 - k, -k + n + 2], [2], -1)):
seq(seq(T(n,k), k=1..n), n=1..10); # Peter Luschny, Sep 18 2018
  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208341 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210554 *)
  • PARI
    T(n,k)={sum(i=1, k, binomial(k-1, i-1)*binomial(n-k+i, i))} \\ Andrew Howroyd, Sep 18 2018

Formula

u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = Sum_{i=1..k} binomial(k-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
T(n,k) = (n - k + 1)*hypergeom([1 - k, n - k + 2], [2], -1). - Peter Luschny, Sep 18 2018

Extensions

Example corrected by Philippe Deléham, Mar 23 2012

A303976 Number of different aperiodic multisets that fit within some normal multiset of size n.

Original entry on oeis.org

1, 3, 9, 26, 75, 207, 565, 1518, 4044, 10703, 28234, 74277, 195103, 511902, 1342147, 3517239, 9214412, 24134528, 63204417, 165505811, 433361425, 1134664831, 2970787794, 7777975396, 20363634815, 53313819160, 139579420528, 365427311171, 956707667616, 2504704955181
Offset: 1

Views

Author

Gus Wiseman, May 03 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.

Examples

			The a(4) = 26 aperiodic multisets:
(1), (2), (3), (4),
(12), (13), (14), (23), (24), (34),
(112), (113), (122), (123), (124), (133), (134), (223), (233), (234),
(1112), (1123), (1222), (1223), (1233), (1234).

Links

Crossrefs

Row sums of A303974.
Cf. A000740, A000837, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&]],{n,10}]
  • PARI
    seq(n)={Vec(sum(d=1, n, moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Feb 04 2021

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
G.f.: Sum_{d>=1} mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)). - Andrew Howroyd, Feb 04 2021

Extensions

Terms a(13) and beyond from Andrew Howroyd, Sep 18 2018

A304623 Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 11, 8, 1, 10, 21, 27, 16, 1, 12, 38, 61, 63, 32, 1, 18, 57, 120, 162, 143, 64, 1, 22, 87, 205, 347, 409, 319, 128, 1, 28, 122, 333, 651, 950, 1000, 703, 256, 1, 32, 164, 506, 1132, 1926, 2504, 2391, 1535, 512, 1, 42, 217, 734, 1840
Offset: 1

Views

Author

Gus Wiseman, May 15 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers, and is aperiodic if its multiplicities are relatively prime.

Examples

			Triangle begins:
1
1    2
1    4    4
1    6   11    8
1   10   21   27   16
1   12   38   61   63   32
1   18   57  120  162  143   64
1   22   87  205  347  409  319  128
The a(4,3) = 11 multisets are (3), (13), (23), (113), (123), (133), (223), (233), (1123), (1223), (1233).

Links

Crossrefs

Row sums are A303976.
Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945, A303974.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]===1&],Max],{n,10}]
  • PARI
    T(n,k) = sum(j=1, n, sumdiv(j, d, sum(i=max(1, j+k-n), d, moebius(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1)))) \\ Andrew Howroyd, Jan 20 2023

Formula

T(n,k) = Sum_{j=1..n} Sum_{d|j} Sum_{i=max(1, j+k-n)..d} mu(j/d)*binomial(k-1, i-1)*binomial(d-1, i-1). - Andrew Howroyd, Jan 20 2023

A304648 Number of different periodic multisets that fit within some normal multiset of weight n.

Original entry on oeis.org

0, 1, 3, 7, 13, 25, 44, 78, 136, 242, 422, 747, 1314, 2326, 4121, 7338, 13052, 23288, 41568, 74329, 133011, 238338, 427278, 766652, 1376258, 2472012, 4441916, 7984990, 14358424, 25826779, 46465956, 83616962, 150497816, 270917035, 487753034, 878244512
Offset: 1

Views

Author

Gus Wiseman, May 15 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers. It is periodic if its multiplicities have a common divisor greater than 1.

Examples

			The a(5) = 13 periodic multisets:
(11), (22), (33), (44),
(111), (222), (333),
(1111), (1122), (1133), (2222), (2233),
(11111).

Links

Crossrefs

Cf. A000217, A001597, A018783, A027941, A178472, A210554, A303547, A303709, A303974, A303976.

Programs

  • Mathematica
    allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]>1&]],{n,10}]
  • PARI
    seq(n)=Vec(sum(d=2, n, -moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x), -n) \\ Andrew Howroyd, Feb 04 2021

Formula

From Andrew Howroyd, Feb 04 2021: (Start)
a(n) = A027941(n) - A303976(n).
G.f.: Sum_{d>=2} -mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)).
(End)

Extensions

a(12)-a(13) from Robert Price, Sep 15 2018
Terms a(14) and beyond from Andrew Howroyd, Feb 04 2021
Showing 1-4 of 4 results.

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