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.

A001277 Number of permutations of length n by rises.

Original entry on oeis.org

1, 3, 12, 56, 321, 2175, 17008, 150504, 1485465, 16170035, 192384876, 2483177808, 34554278857, 515620794591, 8212685046336, 139062777326000, 2494364438359953, 47245095998005059, 942259727190907180, 19737566982241851720, 433234326593362631601, 9943659797649140568863
Offset: 2

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Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A173184.

Formula

Apparently a(n) = A173184(n)-1, partial sums of A000166. - Ralf Stephan, May 23 2004
a(n) = A136123(n,1). Emeric Deutsch and Vladeta Jovovic, Dec 17 2007
Conjecture: a(n) = A177265(n) if n even, = A177265(n)-1 if n odd. - R. J. Mathar, Jun 27 2022
Conjecture: D-finite with recurrence a(n) -n*a(n-1) +(n-1)*a(n-3)=0. - R. J. Mathar, Jul 01 2022

A258817 a(n) = (!0 + !1 +... + !(n-1)) mod n.

Original entry on oeis.org

0, 1, 2, 0, 3, 3, 0, 0, 8, 5, 4, 0, 9, 7, 8, 0, 13, 9, 4, 0, 14, 11, 14, 0, 3, 13, 17, 0, 25, 15, 4, 0, 26, 17, 28, 0, 30, 19, 35, 0, 4, 21, 9, 0, 8, 23, 32, 0, 7, 25, 47, 0, 30, 27, 48, 0, 23, 29, 45, 0, 48, 31, 35, 0, 48, 33, 12, 0, 14, 35, 7, 0, 34, 37, 53
Offset: 1

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Author

Michel Lagneau, Jun 11 2015

Keywords

Comments

!n is a subfactorial number (A000166).
Property of the sequence: a(1) = a(7) = 0 and a(4k) = 0 for k=1,2,...

Examples

			a(5)= 3 because !0 + !1 + !2 + !3 + !4 = 1 + 0 + 1 + 2 + 9 = 13 == 3 mod 5.

Crossrefs

Cf. A000166, A173184.

Programs

  • Maple
    A:= proc(n) option remember; if n<=1 then 1-n else (n-1)*(procname(n-1)+procname(n-2)); fi; end;
a:=n->n!*sum((-1)^k/k!, k=0..n):
lf:=n->add(A(k), k=0..n-1);[seq(lf(n) mod n, n=1..40)];
  • Mathematica
    Table[Mod[Total[Subfactorial[Range[0, n-1]]], n], {n, Range[80]}]

Formula

a(n)= A173184(n) mod n.

A258818 a(n) = (!0 + !1 + ... + !(p-1)) mod p, where p = prime(n).

Original entry on oeis.org

1, 2, 3, 0, 4, 9, 13, 4, 14, 25, 4, 30, 4, 9, 32, 30, 45, 48, 12, 7, 34, 74, 40, 76, 96, 57, 64, 90, 89, 50, 117, 87, 29, 46, 108, 113, 10, 70, 111, 150, 14, 153, 119, 26, 81, 78, 112, 209, 173, 177, 186, 126, 26, 25, 60, 74, 23, 27, 138, 49, 72, 211, 252, 169
Offset: 1

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Author

Michel Lagneau, Jun 11 2015

Keywords

Comments

!n is a subfactorial number (A000166).
This is A173184(p) mod p where p = prime(n) .

Examples

			For n=3, prime(3) = 5 => !0 + !1 + !2 + !3 + !4 = 1 + 0 + 1 + 2 + 9 = 13 == 3 (mod 5), so a(3) = 3.

Links

Crossrefs

Cf. A258817.

Programs

  • Maple
    A:= proc(n) option remember; if n<=1 then 1-n else (n-1)*(procname(n-1)+procname(n-2)); fi; end;
a:=n->n!*sum((-1)^k/k!, k=0..n):
lf:=n->add(A(k), k=0..n-1); [seq(lf(ithprime(n)) mod ithprime(n), n=1..40)];
  • Mathematica
    Table[Mod[Total[Subfactorial[Range[0, n-1]]], n], {n, Prime[Range[70]]}]

A348482 Triangle read by rows: T(n,k) = (Sum_{i=k..n} i!)/(k!) for 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 10, 9, 4, 1, 34, 33, 16, 5, 1, 154, 153, 76, 25, 6, 1, 874, 873, 436, 145, 36, 7, 1, 5914, 5913, 2956, 985, 246, 49, 8, 1, 46234, 46233, 23116, 7705, 1926, 385, 64, 9, 1, 409114, 409113, 204556, 68185, 17046, 3409, 568, 81, 10, 1
Offset: 0

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Author

Werner Schulte, Oct 20 2021

Keywords

Comments

The matrix inverse M = T^(-1) has terms M(n,n) = 1 for n >= 0, M(n,n-1) = -(n+1) for n > 0, and M(n,n-2) = n for n > 1, otherwise 0.

Examples

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :       0       1       2      3      4     5    6   7   8  9
=================================================================
  0 :       1
  1 :       2       1
  2 :       4       3       1
  3 :      10       9       4      1
  4 :      34      33      16      5      1
  5 :     154     153      76     25      6     1
  6 :     874     873     436    145     36     7    1
  7 :    5914    5913    2956    985    246    49    8   1
  8 :   46234   46233   23116   7705   1926   385   64   9   1
  9 :  409114  409113  204556  68185  17046  3409  568  81  10  1
  etc.

Links

Crossrefs

Cf. A109398, A094587, A002104 (row sums), A173184 (alt. row sums), A000012 (main diagonal), A000027(1st subdiagonal), A000290 (2nd subdiagonal), A081437 (3rd subdiagonal), A192398 (4th subdiagonal), A003422 (column 0), A007489 (column 1), A345889 (column 2), A143122.

Programs

  • Mathematica
    T[n_, k_] := Sum[i!, {i, k, n}]/k!; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Oct 20 2021 *)

Formula

T(n,n) = 1 and T(2*n,n) = A109398(n) for n >= 0; T(n,n-1) = n+1 for n > 0; T(n,n-2) = n^2 for n > 1.
T(n,k) - T(n-1,k) = (n!) / (k!) = A094587(n,k) for 0 <= k < n.
T(n,k) = (k+2) * (T(n,k+1) - T(n,k+2)) for 0 <= k < n-1.
T(n,k) = (T(n,k-1) - 1) / k for 0 < k <= n.
T(n,k) * T(n-1,k-1) - T(n-1,k) * T(n,k-1) = (n!) / (k!) for 0 < k < n.
T(n,1) = T(n,0)-1 = Sum_{k=0..n-1} T(n,k)/(k+2) for n > 0 (conjectured).
Sum_{k=0..n} binomial(k+r,k) * (1-k) * T(n+r,k+r) = binomial(n+r+1,n) for n >= 0 and r >= 0.
Sum_{k=0..n} (-1)^k * (k+1) * T(n,k) = (1 + (-1)^n) / 2 for n >= 0.
Sum_{k=0..n} (-1)^k * (k!) * T(n,k) = Sum_{k=0..n} (k!) * (1+(-1)^k) / 2 for n >= 0.
The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k for n >= 0 satisfy the following equations:
(a) p(n,x) - p'(n,x) = (x^(n+1)-1) / (x-1) for n >= 0, where p' is the first derivative of p;
(b) p(n,x) - (n+1) * p(n-1,x) + n * p(n-2,x) = x^n for n > 1.
(c) p(n,x) = (x+1) * p(n-1,x) + 1 + Sum_{i=1..n-1} (d/dx)^i p(n-1,x) for n > 0 (conjectured).
Row sums p(n,1) equal A002104(n+1) for n >= 0.
Alternating row sums p(n,-1) equal A173184(n) for n >= 0 (conjectured).
The three conjectures stated above are true. See links. - Sela Fried, Jul 11 2024.
From Peter Luschny, Jul 11 2024: (Start)
T(n, k) = (t(k) - t(n + 1)) / k!, where t(n) = (-1)^(n + 1) * Gamma(n + 1) * Subfactorial(-(n + 1)).
T(n, k) = A143122(n, k) / k!. (End)
Showing 1-4 of 4 results.

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

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