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 1-8 of 8 results.

A210545 T(n,k) = number of arrays of n nonnegative integers with value i>0 appearing only after i-1 has appeared at least k times.

Original entry on oeis.org

1, 1, 2, 1, 1, 5, 1, 1, 2, 15, 1, 1, 1, 4, 52, 1, 1, 1, 2, 9, 203, 1, 1, 1, 1, 4, 23, 877, 1, 1, 1, 1, 2, 8, 65, 4140, 1, 1, 1, 1, 1, 4, 17, 199, 21147, 1, 1, 1, 1, 1, 2, 8, 40, 654, 115975, 1, 1, 1, 1, 1, 1, 4, 16, 104, 2296, 678570, 1, 1, 1, 1, 1, 1, 2, 8, 33, 291, 8569, 4213597, 1, 1, 1, 1
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2012

Keywords

Examples

			Some solutions for n=13 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....1....1....0....0....1....0....0....1....1....0....1....1
..0....0....1....1....1....1....1....0....0....0....1....1....0....1....1....0
..1....0....0....1....1....1....0....1....1....0....0....0....0....1....1....0
..0....1....1....0....1....1....0....1....0....0....1....0....0....1....1....1
..1....1....0....0....1....2....1....1....0....1....1....0....0....1....2....1
..0....1....0....1....0....1....1....1....1....0....1....1....1....1....2....1
..0....1....0....2....1....0....0....1....1....0....2....0....1....1....2....1
..1....2....0....1....1....0....1....0....0....1....1....0....0....2....1....0
..0....2....1....0....2....0....0....0....2....1....2....1....0....0....0....1
Table starts
..........1.......1......1.....1....1...1...1
..........2.......1......1.....1....1...1...1
..........5.......2......1.....1....1...1...1
.........15.......4......2.....1....1...1...1
.........52.......9......4.....2....1...1...1
........203......23......8.....4....2...1...1
........877......65.....17.....8....4...2...1
.......4140.....199.....40....16....8...4...2
......21147.....654....104....33...16...8...4
.....115975....2296....291....73...32..16...8
.....678570....8569....857...177...65..32..16
....4213597...33825...2634...467..138..64..32
...27644437..140581...8455..1309..315.129..64
..190899322..612933..28424..3813..782.267.128
.1382958545.2795182.100117.11409.2090.582.257

Links

Crossrefs

Cf. A000110 (column 1), A007476 (column 2), A210540 (column 3).

Formula

T(n,k)=1 if n<=k else Sum_{i=0..n-k} binomial(n-k,i)*T(i,k). Proved by R. J. Mathar in the Sequence Fans Mailing List.

A344490 a(n) = 1 + Sum_{k=0..n-3} binomial(n-2,k) * a(k).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 20, 57, 171, 548, 1894, 6998, 27368, 112653, 486645, 2201162, 10397944, 51161168, 261571460, 1386846249, 7612315023, 43190917004, 252951090586, 1527112817054, 9492126182336, 60677428545165, 398489257136529, 2686088269505042, 18567557376240748
Offset: 0

Views

Author

Ilya Gutkovskiy, May 21 2021

Keywords

Crossrefs

Cf. A000629, A210540, A344489, A344491, A344492, A344493.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k] , {k, 0, n - 3}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = (1 + x^2 A[x/(1 - x)])/((1 - x) (1 + x^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = (1 + x^2 * A(x/(1 - x))) / ((1 - x) * (1 + x^2)).

A362549 Number of partitions of [n] whose blocks can be ordered such that the i-th block (except possibly the last) has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 64, 187, 566, 1777, 5820, 19944, 71343, 264719, 1011292, 3953381, 15756609, 63945484, 264384828, 1115246518, 4806957739, 21189601861, 95516470253, 439777682222, 2064164172616, 9853934668051, 47736608806520, 234235866539512, 1162618720397931
Offset: 0

Views

Author

Alois P. Heinz, Apr 24 2023

Keywords

Examples

			a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 9: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 1|23|4.
a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45.
a(6) = 64: 123456, 12345|6, 12346|5, 1234|56, 12356|4, ..., 1|2356|4, 1|235|46, 16|23|45, 1|236|45, 1|23|456.

Links

Crossrefs

Cf. A000110, A007476, A092920, A210540, A362635, A362637, A362893.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n<=t, 1,
      add(b(j, t+1)*binomial(n-t, j), j=0..n-t))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);

A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 6, 11, 23, 60, 179, 553, 1716, 5415, 17801, 61956, 228391, 882309, 3530322, 14531621, 61454091, 267479778, 1200680113, 5561767211, 26553471186, 130366882251, 656668581417, 3387887246292, 17886582294921, 96603394562849, 533645344137390, 3014295344076655
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

Links

Crossrefs

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346051, A346052.

Programs

  • Mathematica
    nmax = 31; A[] = 0; Do[A[x] = x + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]
  • SageMath
    @CachedFunction
def a(n): # a = A346050
    if (n<3): return (0,1,1)[n]
    else: return sum(binomial(n-3,k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 28 2022

Formula

a(0) = 0, a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k).

A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 5, 12, 28, 68, 181, 531, 1671, 5491, 18627, 65299, 237880, 903907, 3580619, 14729777, 62639952, 274442521, 1236730244, 5729809348, 27292248240, 133614280479, 671803041553, 3464970976743, 18309428363425, 99010800275743, 547462187824465, 3093329527120022
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

Links

Crossrefs

Cf. A000994, A000995, A000996, A000997, A000998.
Cf. A007476, A210540, A346050, A346052.

Programs

  • Magma
    function a(n)
  if n lt 3 then return (1+(-1)^n)/2;
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022
  • Mathematica
    nmax = 31; A[] = 0; Do[A[x] = 1 + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]
  • SageMath
    @CachedFunction
def a(n): # a = A346051
    if (n<3): return (1, 0, 1)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

Formula

a(0) = 1, a(1) = 0, a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k).

A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 5, 11, 29, 80, 222, 630, 1881, 6004, 20420, 72979, 270659, 1035590, 4087205, 16675630, 70440641, 307933393, 1390117953, 6462787357, 30871458702, 151298796000, 760250325004, 3915477534861, 20662363081756, 111662169790416, 617482470676567, 3490973387652861
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

Links

Crossrefs

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346050, A346051.

Programs

  • Magma
    function a(n) // a = A346052
  if n lt 3 then return Floor((3-n)/2);
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022
  • Mathematica
    nmax = 31; A[] = 0; Do[A[x] = 1 + x + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]
  • SageMath
    @CachedFunction
def a(n): # a = A346052
    if (n<3): return (1, 1, 0)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

Formula

a(0) = a(1) = 1, a(2) = 0; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k).

A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2*x)) / (1 - 2*x).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 27, 83, 271, 971, 3865, 16879, 78985, 388385, 1987201, 10561385, 58443891, 337724057, 2040085491, 12862712499, 84357800063, 573182197539, 4021203303593, 29062345301487, 216129411635057, 1653180368063361, 13003920016983361, 105158133803473329
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2022

Keywords

Comments

Shifts 3 places left under 2nd-order binomial transform.

Crossrefs

Cf. A000996, A000998, A004211, A007472, A210540, A351343, A351344, A351345.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Binomial[n - 3, k] 2^k a[n - k - 3], {k, 0, n - 3}]]; Table[a[n], {n, 0, 27}]

Formula

a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * 2^k * a(n-k-3).

A351660 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - x)) / (1 - x)^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 15, 33, 81, 225, 679, 2139, 6931, 23185, 80809, 295141, 1128487, 4492363, 18506923, 78584193, 343414489, 1544535129, 7151822771, 34086446307, 167058478355, 840700482197, 4337529697349, 22915761303125, 123863743341203, 684588061704611, 3867278506969535
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 16 2022

Keywords

Crossrefs

Cf. A040027, A210540, A351437, A351707.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Binomial[n - 2, k + 1] a[k], {k, 0, n - 3}]]; Table[a[n], {n, 0, 30}]

Formula

a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-2,k+1) * a(k).
Showing 1-8 of 8 results.

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