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 1-9 of 9 results.

A322773 Column 2 of array in A322770.

Original entry on oeis.org

5, 18, 70, 299, 1393, 7023, 38043, 220054, 1352082, 8784991, 60125371, 432001747, 3248914361, 25508188118, 208592396802, 1772921926183, 15632838989393, 142759592985079, 1348095912827295, 13145321614286610, 132188675368994446, 1369216940917868547
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Links

Crossrefs

Cf. A322764, A322770.
Column k=2 of A346517.

A322774 Column 3 of array in A322770.

Original entry on oeis.org

40, 172, 801, 4025, 21709, 124997, 764538, 4945866, 33710579, 241273791, 1807949285, 14146621349, 115316563400, 977216138500, 8592652709041, 78263082518169, 737228862573509, 7172071557558805, 71964172085006666, 743866850349092130, 7912230538914051723
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Links

Crossrefs

Cf. A322768, A322770.
Column k=3 of A346517.

A322771 Row 1 of array in A322770.

Original entry on oeis.org

1, 3, 18, 172, 2295, 40043, 875936, 23308546, 737478487, 27252363585, 1159335917625, 56103737161197, 3057787510932485, 186102920689311261, 12555513437042340449, 932964243520888524391, 75926403820972271325522, 6733532223196893844825456, 647846856775383975668238328, 67349524752043243630964385000
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Links

Crossrefs

Cf. A322770, A346517.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n A(n, n+1):
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 21 2021
  • Mathematica
    Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
a[n_] := Q[1, n];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Apr 29 2022 *)

Formula

a(n) = A346517(n,n+1) = A346517(n+1,n). - Alois P. Heinz, Jul 21 2021

A322772 Row 2 of array in A322770.

Original entry on oeis.org

2, 9, 70, 801, 12347, 243235, 5908978, 172449180, 5925731200, 235946129714, 10745098631229, 553630279110396, 31978001903989065, 2054387367168242251, 145795148420558536232, 11361381129471379493270, 967044630942570464100761, 89483154423059719127570924, 8963545185499520505954151682
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Links

Crossrefs

Cf. A322770, A346517.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n A(n, n+2):
seq(a(n), n=0..18);  # Alois P. Heinz, Jul 21 2021
  • Mathematica
    Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
a[n_] := Q[2, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Apr 29 2022 *)

Formula

a(n) = A346517(n,n+2) = A346517(n+2,n). - Alois P. Heinz, Jul 21 2021

A322775 Main diagonal of array in A322770.

Original entry on oeis.org

1, 3, 70, 4025, 431636, 75114998, 19449364539, 7059006496410, 3437740821764782, 2172596362927904998, 1735323511562307096513, 1714589657484702134799407, 2058734042458059390066278968, 2959366820014712815406422598389, 5028070020338050558675966002870476
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Crossrefs

Cf. A322770.

A346517 Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k} into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 3, 3, 5, 15, 9, 5, 9, 15, 52, 31, 18, 18, 31, 52, 203, 120, 70, 40, 70, 120, 203, 877, 514, 299, 172, 172, 299, 514, 877, 4140, 2407, 1393, 801, 457, 801, 1393, 2407, 4140, 21147, 12205, 7023, 4025, 2295, 2295, 4025, 7023, 12205, 21147
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2021

Keywords

Comments

Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i) into distinct factors; A(2,2) = 5: 2*3*6, 4*9, 3*12, 2*18, 36.

Examples

			A(2,2) = 5: 1122, 11|22, 1|122, 112|2, 1|12|2.
Square array A(n,k) begins:
    1,    1,    2,     5,    15,     52,    203,     877, ...
    1,    1,    3,     9,    31,    120,    514,    2407, ...
    2,    3,    5,    18,    70,    299,   1393,    7023, ...
    5,    9,   18,    40,   172,    801,   4025,   21709, ...
   15,   31,   70,   172,   457,   2295,  12347,   70843, ...
   52,  120,  299,   801,  2295,   6995,  40043,  243235, ...
  203,  514, 1393,  4025, 12347,  40043, 136771,  875936, ...
  877, 2407, 7023, 21709, 70843, 243235, 875936, 3299218, ...
  ...

Links

Crossrefs

Columns (or rows) k=0-10 give: A000110, A087648, A322773, A322774, A346897, A346898, A346899, A346900, A346901, A346902, A346903.
Main diagonal gives A094574.
First upper (or lower) diagonal gives A322771.
Second upper (or lower) diagonal gives A322772.
Antidiagonal sums give A346518.
Cf. A002110, A045778, A322770, A346500.

Programs

  • Maple
    g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
A:= (n, k)-> g(p(n)*p(k)$2):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n
  • Mathematica
    (* Q is A322770 *)
Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
A[n_, k_] := Q[Abs[n-k], Min[n, k]];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Aug 19 2021 *)

Formula

A(n,k) = A045778(A002110(n)*A002110(k)).
A(n,k) = A(k,n).
A(n,k) = A322770(abs(n-k),min(n,k)).

A322765 Array read by upwards antidiagonals: T(m,n) = number of set partitions of the multiset consisting of one copy each of x_1, x_2, ..., x_m, and two copies each of y_1, y_2, ..., y_n, for m >= 0, n >= 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 9, 5, 11, 26, 66, 15, 36, 92, 249, 712, 52, 135, 371, 1075, 3274, 10457, 203, 566, 1663, 5133, 16601, 56135, 198091, 877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, 4140, 13082, 43263, 149410, 537813, 2014321, 7837862, 31638625, 132315780
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Examples

			The array begins:
    1,    2,     9,     66,      712,     10457,      198091, ...
    1,    4,    26,    249,     3274,     56135,     1207433, ...
    2,   11,    92,   1075,    16601,    325269,     7837862, ...
    5,   36,   371,   5133,    91226,   2014321,    53840640, ...
   15,  135,  1663,  26683,   537813,  13241402,   389498179, ...
   52,  566,  8155, 149410,  3376696,  91914202,  2955909119, ...
  203, 2610, 43263, 894124, 22451030, 670867539, 23456071495, ...
  ...

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.

Links

Crossrefs

Rows include A020555, A322766, A322767.
Columns include A000110, A035098, A322764, A322768.
Main diagonal is A322769.
See A322770 for partitions into distinct parts.
Cf. A001055, A002110, A346500.

Programs

  • Maple
    B := n -> combinat[bell](n):
P := proc(m,n) local k; global B; option remember;
if n = 0 then B(m)  else
(1/2)*( P(m+2,n-1) + P(m+1,n-1) + add( binomial(n-1,k)*P(m,k), k=0..n-1) ); fi; end; # P(m,n) (which is Knuth's notation) is T(m,n)
  • Mathematica
    P[m_, n_] := P[m, n] = If[n == 0, BellB[m], (1/2)(P[m+2, n-1] + P[m+1, n-1] + Sum[Binomial[n-1, k] P[m, k], {k, 0, n-1}])];
Table[P[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 02 2019, from Maple *)
  • PARI
    {T(n, k) = if(k==0, sum(j=0, n, stirling(n, j, 2)), (T(n+2, k-1)+T(n+1, k-1)+sum(j=0, k-1, binomial(k-1, j)*T(n, j)))/2)} \\ Seiichi Manyama, Nov 21 2020

Formula

Knuth p. 779 gives a recurrence using the Bell numbers A000110 (see Maple program).
From Alois P. Heinz, Jul 21 2021: (Start)
A(n,k) = A001055(A002110(n+k)*A002110(k)).
A(n,k) = A346500(n+k,k). (End)

A087648 a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).

Original entry on oeis.org

1, 3, 9, 31, 120, 514, 2407, 12205, 66491, 386699, 2388096, 15589732, 107165081, 773106715, 5836100685, 45981026703, 377230766908, 3215977070706, 28437411817135, 260380616093533, 2464930698184351, 24091925888687459, 242802079705721156, 2520198597834860148
Offset: 0

Views

Author

Vladeta Jovovic, Sep 23 2003

Keywords

Comments

Sum of last number in all set partitions of n+1. E.g. The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}, so a(2) = 1+2+1+2+3 = 9. - Franklin T. Adams-Watters, Jun 07 2006
Number of partitions of the (n+2)-multiset {0,0,1,2,...,n} into distinct multisets. Also number of factorizations of 2 * Product_{i=1..n+1} prime(i) into distinct factors. - Alois P. Heinz, Jul 30 2021

Links

Crossrefs

Cf. A000110, A035098, A059606.
Main diagonal of A120057, row sums of A120095.
Column 1 of array in A322770.
Row n=2 of A346520.

Programs

  • Magma
    [(1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)) : n in [0..30]]; // Vincenzo Librandi, Nov 13 2011
  • Mathematica
    f[0]=1; f[n_] := Sum[ StirlingS2[n, k]*Binomial[k+2, k ], {k, 1, n}]; Table[ f[n], {n, 0, 20}] (* Zerinvary Lajos, Mar 31 2007 *)
(#[[3]]+#[[2]]-#[[1]])/2&/@Partition[BellB[Range[0,30]],3,1] (* Harvey P. Dale, Jul 20 2021 *)

A346518 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} into distinct multisets for 0 <= j <= n.

Original entry on oeis.org

1, 2, 5, 16, 53, 202, 826, 3724, 17939, 93390, 516125, 3042412, 18923139, 124368810, 857827458, 6208594458, 46937360868, 370335617694, 3039823038753, 25928519847988, 229285625745624, 2099543718917418, 19872430464012935, 194203934113959970, 1956736801957704866
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2021

Keywords

Comments

Also total number of factorizations of Product_{i=1..n-j} prime(i) * Product_{i=1..j} prime(i) into distinct factors for 0 <= j <= n.

Links

Crossrefs

Antidiagonal sums of A346517.
Cf. A002110, A045778, A346490.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n add(A(n-j, j), j=0..n):
seq(a(n), n=0..24);
  • Mathematica
    (* Q is A322770 *)
Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2) (Q[m + 2, n - 1] +
     Q[m + 1, n - 1] - Sum[Binomial[n - 1, k] Q[m, k], {k, 0, n - 1}])];
A[n_, k_] := Q[Abs[n - k], Min[n, k]];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 06 2022 *)

Formula

a(n) = Sum_{j=0..n} A045778(A002110(n-j)*A002110(j)).
a(n) = Sum_{j=0..n} A346517(n-j,j).
Showing 1-9 of 9 results.

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

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