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 1-7 of 7 results.

A179807 Antidiagonal sums of A175105.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 17, 34, 71, 153, 337, 755, 1713, 3925, 9064, 21065, 49213, 115477, 271961, 642495, 1521899, 3613189, 8595037, 20480600, 48874131, 116781789, 279358063, 668925263, 1603145325, 3845056065, 9228442541, 22162387821
Offset: 1

Views

Author

Mats Granvik, Jul 28 2010

Keywords

Comments

The ratio a(n+1)/a(n) tends to 1 + sqrt(2), A014176.

Crossrefs

Cf. A175105, A002193.

A176084 Row sums of A175105.

Original entry on oeis.org

1, 2, 4, 9, 22, 58, 160, 454, 1311, 3828, 11260, 33290, 98778, 293866, 875960, 2614891, 7814544, 23373354, 69955372, 209478678, 627521578, 1880400340, 5636065932, 16895989570, 50658893073
Offset: 1

Views

Author

Mats Granvik, Apr 08 2010

Keywords

Comments

a(n+1)/a(n) tends to 3.
a(n)/A007051(n) tends to Product_{k>=1} (1-1/((3^k + 1)/2)). To observe the asymptote one needs 1000 or more decimal digits of the constant c=Product_{k>=1} (1-1/((3^k + 1)/2)). - Mats Granvik, Jan 02 2015

Examples

			a(25)/A007051(24) = 50658893073/141214768241 = 0.35873650967258963431... which is close to 0.35792312728995990302591...

Crossrefs

Cf. A177510.

Programs

  • Mathematica
    Clear[t]; nn = 25; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] + Sum[t[n - i, k], {i, 1, k - 1}], 0]; Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}](* Mats Granvik, Jan 02 2015 *)

Formula

a(n) ~ Product_{k>=1} (1-1/((3^k + 1)/2))*A007051(n). - Mats Granvik, Jan 01 2015

A179815 Numbers such that n=2*k in triangle A175105.

Original entry on oeis.org

1, 3, 21, 163, 1259, 9657, 73949, 566797, 4352755, 33501979, 258431853, 1997743677, 15473296249
Offset: 1

Views

Author

Mats Granvik, Jul 28 2010

Keywords

Comments

All numbers in this sequence are odd.

A186268 Central coefficients of A175105.

Original entry on oeis.org

1, 2, 10, 64, 442, 3166, 23164, 171884, 1288404, 9731818, 73951550, 564689040, 4329242164, 33302459936, 256913334792, 1986880768892, 15399049701698
Offset: 1

Views

Author

Mats Granvik, Feb 16 2011

Keywords

A051731 Triangle read by rows: T(n, k) = 1 if k divides n, T(n, k) = 0 otherwise, for 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Keywords

Comments

T(n, k) is the number of partitions of n into k equal parts. - Omar E. Pol, Apr 21 2018
This triangle is the lower triangular array L in the LU decomposition of the square array A003989. - Peter Bala, Oct 15 2023

Examples

			The triangle T(n, k) begins:
  n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  1:  1
  2:  1  1
  3:  1  0  1
  4:  1  1  0  1
  5:  1  0  0  0  1
  6:  1  1  1  0  0  1
  7:  1  0  0  0  0  0  1
  8:  1  1  0  1  0  0  0  1
  9:  1  0  1  0  0  0  0  0  1
  10: 1  1  0  0  1  0  0  0  0  1
  11: 1  0  0  0  0  0  0  0  0  0  1
  12: 1  1  1  1  0  1  0  0  0  0  0  1
  13: 1  0  0  0  0  0  0  0  0  0  0  0  1
  14: 1  1  0  0  0  0  1  0  0  0  0  0  0  1
  15: 1  0  1  0  1  0  0  0  0  0  0  0  0  0  1
  ... Reformatted and extended. - _Wolfdieter Lang_, Nov 12 2014

Links

Crossrefs

Cf. A000007, A000203, A002260, A003989, A048158, A049820, A054525.
Cf. A074854, A101508, A115361, A129372, A172119, A175105.
Cf. A000005 (row sums), A032741(n+2) (diagonal sums).
Cf. A243987 (partial sums per row).
Cf. A134546 (A004736 * T, matrix multiplication).
Variants: A113704, A077049, A077051.

Programs

  • Haskell
    a051731 n k = 0 ^ mod n k
a051731_row n = a051731_tabl !! (n-1)
a051731_tabl = map (map a000007) a048158_tabl
-- Reinhard Zumkeller, Aug 13 2013
  • Magma
    [0^(n mod k): k in [1..n], n in [1..17]]; // G. C. Greubel, Jun 22 2024
  • Maple
    A051731 := proc(n, k) if n mod k = 0 then 1 else 0 end if end proc:
# R. J. Mathar, Jul 14 2012
  • Mathematica
    Flatten[Table[If[Mod[n, k] == 0, 1, 0], {n, 20}, {k, n}]]
  • PARI
    for(n=1,17,for(k=1,n,print1(!(n%k)", "))) \\ Charles R Greathouse IV, Mar 14 2012
  • Python
    from math import isqrt, comb
def A051731(n): return int(not (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))%(n-comb(a,2))) # Chai Wah Wu, Nov 13 2024
  • Sage
    A051731_row = lambda n: [int(k.divides(n)) for k in (1..n)]
for n in (1..17): print(A051731_row(n)) # Peter Luschny, Jan 05 2018

Formula

{T(n, k)*k, k=1..n} setminus {0} = divisors(n).
Sum_{k=1..n} T(n, k)*k^i = sigma[i](n), where sigma[i](n) is the sum of the i-th power of the positive divisors of n.
Sum_{k=1..n} T(n, k) = A000005(n).
Sum_{k=1..n} T(n, k)*k = A000203(n).
T(n, k) = T(n-k, k) for k <= n/2, T(n, k) = 0 for n/2 < k <= n-1, T(n, n) = 1.
Rows given by A074854 converted to binary. Example: A074854(4) = 13 = 1101_2; row 4 = 1, 1, 0, 1. - Philippe Deléham, Oct 04 2003
From Paul Barry, Dec 05 2004: (Start)
Binomial transform (product by binomial matrix) is A101508.
Columns have g.f.: x^k/(1-x^(k+1)) (k >= 0). (End)
Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/k) if k|n, 0 otherwise. - Paul D. Hanna, Jan 09 2006
From Gary W. Adamson, Apr 15 2007, May 10 2007: (Start)
Equals A129372 * A115361 as infinite lower triangular matrices.
A054525 is the inverse of this triangle (as lower triangular matrix).
This triangle * [1, 2, 3, ...] = sigma(n) (A000203).
This triangle * [1/1, 1/2, 1/3, ...] = sigma(n)/n. (End)
From Reinhard Zumkeller, Nov 01 2009: (Start)
T(n, k) = 0^(n mod k).
T(n, k) = A000007(A048158(n, k)). (End)
From Mats Granvik, Jan 26 2010, Feb 10 2010, Feb 16 2010: (Start)
T(n, k) = A172119(n) mod 2.
T(n, k) = A175105(n) mod 2.
T(n, k) = Sum_{i=1..k-1} (T(n-i, k-1) - T(n-i, k)) for k > 1 and T(n, 1) = 1.
(Jeffrey O. Shallit kindly provided a clarification along with a proof of this formula.) (End)
A049820(n) = number of zeros in n-th row. - Reinhard Zumkeller, Mar 09 2010
The determinant of this matrix where T(n, n) has been swapped with T(1,k) is equal to the n-th term of the Mobius function. - Mats Granvik, Jul 21 2012
T(n, k) = Sum_{y=1..n} Sum_{x=1..n} [GCD((x/y)*(k/n), n) = k]. - Mats Granvik, Dec 17 2023

Extensions

Edited by Peter Luschny, Oct 18 2023

A177517 Triangle T(n,k) read by rows defined by recurrence T(n,1)=A000007(n-1) and T(n,k) = sum_{i=1..k-1} T(n-i,k-1) if k>1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 1, 5, 4, 1, 0, 0, 0, 0, 6, 9, 5, 1, 0, 0, 0, 0, 5, 15, 14, 6, 1, 0, 0, 0, 0, 3, 20, 29, 20, 7, 1, 0, 0, 0, 0, 1, 22, 49, 49, 27, 8, 1, 0, 0, 0, 0, 0, 20, 71, 98, 76, 35, 9, 1, 0, 0, 0, 0, 0, 15, 90, 169, 174, 111, 44, 10, 1, 0, 0, 0, 0, 0, 9, 101, 259, 343, 285, 155, 54, 11, 1, 0, 0, 0, 0, 0, 4, 101, 359, 602, 628, 440, 209, 65, 12, 1, 0, 0, 0, 0, 0, 1, 90, 455, 961, 1230, 1068, 649, 274, 77, 13, 1
Offset: 1

Views

Author

Mats Granvik, Roger L. Bagula, Dec 11 2010

Keywords

Comments

A008302 is the main entry for this triangle.
Essentially A060701 which is equal to this table beginning from the second column.
The recurrence formula is similar to the recurrence for A177978.

Examples

			1,
0,1,
0,0,1,
0,0,1,1,
0,0,0,2,1,
0,0,0,2,3,1,
0,0,0,1,5,4,1,
0,0,0,0,6,9,5,1,
0,0,0,0,5,15,14,6,1,
0,0,0,0,3,20,29,20,7,1,
0,0,0,0,1,22,49,49,27,8,1

Links

Crossrefs

Cf. A008302, A060701, A177978, A175105. Column sums are A000142. Row sums are A008930.

Programs

  • Mathematica
    t[1, 1] = 1; t[n_, 1] = 0; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0];
Flatten[Table[t[n, k], {n, 12}, {k, n}]]
(* Robert G. Wilson v, Jun 24 2011 *) (* corrected by Mats Granvik, Jan 23 2012 *)

Formula

T(n,k) = A008302(k-2,n-k), n>=k>1. - R. J. Mathar, Dec 15 2010

A179748 Triangle T(n,k) read by rows. T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 5, 4, 1, 1, 1, 2, 6, 9, 5, 1, 1, 1, 2, 6, 15, 14, 6, 1, 1, 1, 2, 6, 20, 29, 20, 7, 1, 1, 1, 2, 6, 23, 49, 49, 27, 8, 1, 1, 1, 2, 6, 24, 71, 98, 76, 35, 9, 1, 1, 1, 2, 6, 24, 91, 169, 174, 111, 44, 10, 1, 1, 1, 2, 6, 24, 106, 259, 343, 285, 155, 54, 11, 1
Offset: 1

Views

Author

Mats Granvik, Jul 26 2010

Keywords

Comments

Recurrence is half of the recurrence for divisibility in A051731. That is, without subtracting (Sum_{i=1..k-1} T(n-i,k)).
Rows tend to factorial numbers.
Row sums are A177510.

Examples

			Triangle begins:
01: 1;
02: 1, 1;
03: 1, 1, 1;
04: 1, 1, 2, 1;
05: 1, 1, 2, 3,  1;
06: 1, 1, 2, 5,  4,   1;
07: 1, 1, 2, 6,  9,   5,   1;
08: 1, 1, 2, 6, 15,  14,   6,    1;
09: 1, 1, 2, 6, 20,  29,  20,    7,    1;
10: 1, 1, 2, 6, 23,  49,  49,   27,    8,    1;
11: 1, 1, 2, 6, 24,  71,  98,   76,   35,    9,    1;
12: 1, 1, 2, 6, 24,  91, 169,  174,  111,   44,   10,    1;
13: 1, 1, 2, 6, 24, 106, 259,  343,  285,  155,   54,   11,    1;
14: 1, 1, 2, 6, 24, 115, 360,  602,  628,  440,  209,   65,   12,   1;
15: 1, 1, 2, 6, 24, 119, 461,  961, 1230, 1068,  649,  274,   77,  13,   1;
16: 1, 1, 2, 6, 24, 120, 551, 1416, 2191, 2298, 1717,  923,  351,  90,  14,  1;
17: 1, 1, 2, 6, 24, 120, 622, 1947, 3606, 4489, 4015, 2640, 1274, 441, 104, 15, 1;
...

Crossrefs

Cf. A175105, A051731, A179749, A179750, A000142.

Programs

  • Sage
    @CachedFunction
def T(n, k): # A179748
    if n == 0:  return int(k==0);
    if k == 1:  return int(n>=1);
    return sum( T(n-i, k-1) for i in [1..k-1] );
for n in [1..15]: print([ T(n, k) for k in [1..n] ])
# Joerg Arndt, Mar 24 2014

Formula

T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).
Showing 1-7 of 7 results.

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