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 1-3 of 3 results.

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 1, 2, 6, 24, 1, 1, 1, 2, 6, 24, 120, 1, 1, 1, 2, 6, 24, 120, 720, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880
Offset: 1

Views

Author

Johannes W. Meijer, Oct 05 2009

Keywords

Examples

			Triangle starts:
1,
1, 1,
1, 1, 1,
1, 1, 1, 2,
1, 1, 1, 2, 6,
1, 1, 1, 2, 6, 24,
1, 1, 1, 2, 6, 24, 120,
1, 1, 1, 2, 6, 24, 120, 720,
1, 1, 1, 2, 6, 24, 120, 720, 5040,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,
...

Crossrefs

Cf. A138771, A165675 and A000142.
A000012 (3x), A007395, A010722, A010863 equal the first six left hand columns.
A159333 equals, for n=>-1, all right hand columns.
A067078 equals the row sums.

Programs

  • Maple
    nmax:=11: for n from 1 to nmax do a(n,1):=1 od: for n from 2 to nmax do for m from 2 to n do a(n,m):=(m-2)! od: od: for n from 1 to nmax do seq(a(n,m),m=1..n) od;

Formula

a(n) = A138771(n)/A165675(n-1).

A165675 Triangle read by rows. T(n, k) = (n - k + 1)! * H(k, n - k), where H are the hyperharmonic numbers. For 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 11, 5, 1, 24, 50, 26, 7, 1, 120, 274, 154, 47, 9, 1, 720, 1764, 1044, 342, 74, 11, 1, 5040, 13068, 8028, 2754, 638, 107, 13, 1, 40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1, 362880, 1026576, 663696, 241128, 60216, 11274, 1650, 191, 17, 1
Offset: 0

Views

Author

Johannes W. Meijer, Oct 05 2009

Keywords

Comments

Previous name: Extended triangle related to the asymptotic expansions of the E(x, m = 2, n).
For the definition of the hyperharmonic numbers see the formula section.
This triangle is the same as triangle A165674 except for the extra left-hand column T(n, 0) = n!. The T(n) formulas for the right-hand columns generate the coefficients of this extra left-hand column.
Leroy Quet discovered triangle A105954 which is the reversal of our triangle.
In square format, row k gives the (n-1)-st elementary symmetric function of {k, k+1, k+2,..., k+n}, as in the Mathematica section. - Clark Kimberling, Dec 29 2011

Examples

			Triangle T(n, k) begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     3,    1;
  [3]    6,    11,    5,    1;
  [4]   24,    50,   26,    7,   1;
  [5]  120,   274,  154,   47,   9,   1;
  [6]  720,  1764, 1044,  342,  74,  11,  1;
  [7] 5040, 13068, 8028, 2754, 638, 107, 13, 1;
Seen as an array (the triangle arises when read by descending antidiagonals):
  [0] 1,  1,   2,    6,    24,    120,     720,     5040, ...
  [1] 1,  3,  11,   50,   274,   1764,   13068,   109584, ...
  [2] 1,  5,  26,  154,  1044,   8028,   69264,   663696, ...
  [3] 1,  7,  47,  342,  2754,  24552,  241128,  2592720, ...
  [4] 1,  9,  74,  638,  5944,  60216,  662640,  7893840, ...
  [5] 1, 11, 107, 1066, 11274, 127860, 1557660, 20355120, ...
  [6] 1, 13, 146, 1650, 19524, 245004, 3272688, 46536624, ...
  [7] 1, 15, 191, 2414, 31594, 434568, 6314664, 97053936, ...

Crossrefs

A105954 is the reversal of this triangle.
A165674, A138771 and A165680 are related triangles.
A080663 equals the third right hand column.
A000142 equals the first left hand column.
A093345 are the row sums.
Columns include A165676, A165677, A165678 and A165679.

Programs

  • Maple
    nmax := 8; for n from 0 to nmax do a(n, 0) := n! od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (n-m+1)*a(n-1, m) + a(n-1, m-1) od: od: seq(seq(a(n, m), m=0..n), n=0..nmax);
# Johannes W. Meijer, revised Nov 27 2012
# Shows the array format, using hyperharmonic numbers.
H := proc(n, k) option remember; if n = 0 then 1/(k+1)
else add(H(n - 1, j), j = 0..k) fi end:
seq(lprint(seq((k + 1)!*H(n, k), k = 0..7)), n = 0..7);
# Shows the array format, using the hypergeometric formula.
A := (n, k) -> (k+1)*((n + k)! / n!)*hypergeom([-k, 1, 1], [2, n + 1], 1):
seq(lprint(seq(simplify(A(n, k)), k = 0..7)), n = 0..7);
# Peter Luschny, Jul 03 2022
  • Mathematica
    a[n_] := SymmetricPolynomial[n - 1, t[n]]; z = 10;
t[n_] := Table[k - 1, {k, 1, n}]; t1 = Table[a[n], {n, 1, z}]  (* A000142 *)
t[n_] := Table[k,     {k, 1, n}]; t2 = Table[a[n], {n, 1, z}]  (* A000254 *)
t[n_] := Table[k + 1, {k, 1, n}]; t3 = Table[a[n], {n, 1, z}]  (* A001705 *)
t[n_] := Table[k + 2, {k, 1, n}]; t4 = Table[a[n], {n, 1, z}]  (* A001711 *)
t[n_] := Table[k + 3, {k, 1, n}]; t5 = Table[a[n], {n, 1, z}]  (* A001716 *)
t[n_] := Table[k + 4, {k, 1, n}]; t6 = Table[a[n], {n, 1, z}]  (* A001721 *)
t[n_] := Table[k + 5, {k, 1, n}]; t7 = Table[a[n], {n, 1, z}]  (* A051524 *)
t[n_] := Table[k + 6, {k, 1, n}]; t8 = Table[a[n], {n, 1, z}]  (* A051545 *)
t[n_] := Table[k + 7, {k, 1, n}]; t9 = Table[a[n], {n, 1, z}]  (* A051560 *)
t[n_] := Table[k + 8, {k, 1, n}]; t10 = Table[a[n], {n, 1, z}] (* A051562 *)
t[n_] := Table[k + 9, {k, 1, n}]; t11 = Table[a[n], {n, 1, z}] (* A051564 *)
t[n_] := Table[k + 10, {k, 1, n}];t12 = Table[a[n], {n, 1, z}] (* A203147 *)
t = {t1, t2, t3, t4, t5, t6, t7, t8, t9, t10};
TableForm[t]  (* A165675 in square format *)
m[i_, j_] := t[[i]][[j]];
(* A165675 as a sequence *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 10}, {i, 1, n}]]
(* Clark Kimberling, Dec 29 2011 *)
A[n_, k_] := (k + 1)*((n + k)! / n!)*HypergeometricPFQ[{-k, 1, 1}, {2, n + 1}, 1];
Table[A[n, k], {n, 0, 7}, {k, 0, 7}] // TableForm (* Peter Luschny, Jul 03 2022 *)
  • Python
    from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0:
        return [1]
    row = Trow(n - 1) + [1]
    for m in range(n - 1, 0, -1):
        row[m] = (n - m + 1) * row[m] + row[m - 1]
    row[0] *= n
    return row
for n in range(9): print(Trow(n))  # Peter Luschny, Feb 27 2025

Formula

The hyperharmonic numbers are H(n, k) = Sum_{j=0..k} H(n - 1, j), with base condition H(0, k) = 1/(k + 1).
T(n, k) = (n - k + 1)*T(n - 1, k) + T(n - 1, k - 1), 1 <= k <= n-1, with T(n, 0) = n! and T(n, n) = 1.
From Peter Luschny, Jul 03 2022: (Start)
The rectangular array is given by:
A(n, k) = (k + 1)!*H(n, k).
A(n, k) = (k + 1)*((n + k)! / n!)*hypergeom([-k, 1, 1], [2, n + 1], 1). (End)
From Werner Schulte, Feb 26 2025: (Start)
T(n, k) = n * T(n-1, k) + (n-1)! / (k-1)! for 0 < k < n.
T(n, k) = (Sum_{i=k..n} 1/i) * n! / (k-1)! for 0 < k <= n.
Matrix inverse M = T^(-1) is given by: M(n, n) = 1, M(n, n-1) = 1 - 2 * n for n > 0, M(n, n-2) = (n-1)^2 for n > 1, and M(i, j) = 0 otherwise. (End)

Extensions

New name from Peter Luschny, Jul 03 2022

A138772 Number of entries in the second cycles of all permutations of {1,2,...,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

Original entry on oeis.org

0, 1, 5, 27, 168, 1200, 9720, 88200, 887040, 9797760, 117936000, 1536796800, 21555072000, 323805081600, 5187108326400, 88268019840000, 1590132031488000, 30233431388160000, 605024315191296000, 12711912992722944000, 279783730940313600000, 6437458713635389440000
Offset: 1

Views

Author

Emeric Deutsch, Apr 10 2008

Keywords

Examples

			a(3) = 5 because the number of entries in the second cycles of (1)(2)(3), (1)(23), (132), (12)(3), (123) and (13)(2) is 1+2+0+1+0+1=5.

Links

Crossrefs

Cf. A138771.
Cf. A159324, A000254.
Column k=2 of A185105.

Programs

  • GAP
    List([1..30], n-> (n-1)*(n+2)*Factorial(n-1)/4); # G. C. Greubel, Jul 07 2019
  • Magma
    [(n-1)*(n+2)*Factorial(n-1)/4: n in [1..30]]; // G. C. Greubel, Jul 07 2019
  • Maple
    seq((1/4)*factorial(n-1)*(n-1)*(n+2), n = 1 .. 30);
  • Mathematica
    Table[((1/4)(n-1)!(n-1)(n+2)),{n,1,30}] (* Vincenzo Librandi, May 14 2012 *)
  • PARI
    vector(30, n, (n-1)*(n+2)*(n-1)!/4) \\ G. C. Greubel, Jul 07 2019
  • Sage
    [(n-1)*(n+2)*factorial(n-1)/4 for n in (1..30)] # G. C. Greubel, Jul 07 2019

Formula

a(n) = (1/4)*(n-1)!*(n-1)*(n+2).
a(n) = (n+1)*a(n-1) + (n-2)!.
a(n) = (n-1)*a(n-1) + n!/2.
a(n) = Sum_{k=0..n-1} k*A138771(n,k).
E.g.f. if offset 0: x*(2-x)/(2*(1-x)^3). Such e.g.f. computations resulted from e-mail exchange with Gary Detlefs. - Wolfdieter Lang, May 27 2010
a(n) = A000254(n-1) + A159324(n-1). - Gary Detlefs, May 13 2012
a(n) = n! * Sum_{i=1..n} (Sum_{j=1..i} (j/i)). - Pedro Caceres, Apr 19 2019
E.g.f.: ( x*(2-x)/(1-x)^2 + 2*log(1-x) )/4. - G. C. Greubel, Jul 07 2019
D-finite with recurrence a(n) +(-n-1)*a(n-1) -2*a(n-2) +2*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
Showing 1-3 of 3 results.

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