A062111 -id:A062111 - cp's OEIS frontend

A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers.

0, 2, 11, 39, 114, 300, 741, 1757, 4052, 9162, 20415, 44979, 98214, 212888, 458633, 982905, 2097000, 4456278, 9436995, 19922735, 41942810, 88080132, 184549101, 385875669, 805306044, 1677721250, 3489660551, 7247756907, 15032385102, 31138512432, 64424508945

Offset: 0

Jean-François Alcover and Francois Alcover, Sep 11 2016

After 0, is this the second column of A108284? [Bruno Berselli, Sep 13 2016 - this comment may be removed if the property is confirmed.]

			Starting from the triangle:
   0,  1,  2,  3,  4,  5, ...
   1,  3,  5,  7,  9, ...
   4,  8, 12, 16, ...
  12, 20, 28, ...
  32, 48, ...
  80, ...
  ...
the first terms are:
a(0) = 0;
a(1) = a(0) + 1 + 1 = 2;
a(2) = a(1) + 4 + 3 + 2 = 11;
a(3) = a(2) + 12 + 8 + 5 + 3 = 39, etc.
First column is A001787: n*2^(n-1).

Index entries for linear recurrences with constant coefficients, signature (7,-19,25,-16,4).

Cf. A001787, A058877, A062111, A152920.

[(2^(n+2)-n-3)*n/2: n in [0..40]]; // Vincenzo Librandi, Sep 13 2016

A276659:=n->n*(2^(n+2) - n - 3)/2: seq(A276659(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

t[0, k_] := k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 1, k + 1]; a[n_] := Sum[t[m, k], {m, 0, n}, {k, 0, n - m}]; Table[a[n], {n, 0, 30}]
Table[(2^(n + 2) - n - 3) n / 2, {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

x='x+O('x^99); concat(0, Vec(x*(2-3*x)/((1-x)^3*(1-2*x)^2))) \\ Altug Alkan, Sep 14 2017

O.g.f.: x*(2 - 3*x)/((1 - x)^3*(1 - 2*x)^2).
E.g.f.: x*exp(x)*(8*exp(x) - x - 4)/2.
a(n) = n*(2^(n+2) - n - 3)/2.
a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5) for n > 4.
a(n) = a(n-1) + A058877(n+1). - R. J. Mathar, Sep 14 2016
a(n) = Sum_{k=2..n+3} Sum_{i=2..n+3} k * C(n-i+3,k). - Wesley Ivan Hurt, Sep 20 2017

Edited and extended by Bruno Berselli, Sep 13 2016

A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k(k + 1 + 2n), k >= 0, n >= 0, read by antidiagonals upwards.

Original entry on oeis.org

1, 4, 3, 12, 8, 5, 32, 20, 12, 7, 80, 48, 28, 16, 9, 192, 112, 64, 36, 20, 11, 448, 256, 144, 80, 44, 24, 13, 1024, 576, 320, 176, 96, 52, 28, 15, 2304, 1280, 704, 384, 208, 112, 60, 32, 17, 5120, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 11264, 6144, 3328, 1792, 960, 512, 272, 144, 76, 40, 21

Offset: 0

Wolfdieter Lang, Oct 03 2019

The array A(k, n) arises from the following Pascal-type triangles PTodd(k), k >= 0 based on the positive odd integers A005408.
For example, the Pascal-type triangle PTodd(k), for k = 3 is
   1     3     5     7
      4     8    12
        12    20
           32
Taken upside-down such triangles become so-called addition towers of height k+1 (Rechenturm in German elementary schools; thanks to my correspondent Bennet D.), starting with any k+1 numbers. Here the positive odd numbers are used.
The sequence s of the final number of these Pascal-type triangles PT(k), for k >= 0, begins 1, 4, 12, 32, ...; s(k) = (k+1)*2^k = A001787(k+1), for k >= 0.
For k -> infinity the left-aligned row sequences build the array A(k, n), with k >= 0 and n >= 0, namely A(k, n) = 2^k*(k + 2*n + 1); this array begins:
  k\n  0   1   2   3   4    5 ...
  -------------------------------
  0:   1   3   5   7   9   11 ... {A005408(n)}
  1:   4   8  12  16  20   24 ... {A008586(n+1)}
  2:  12  20  28  36  44   52 ... {A017113(n+1)}
  3:  32  48  64  80  96  112 ... {A008598(n+2)}
  4:  80 112 144 176 208  240 ... {16*A005408(n+2)}
  5: 192 256 320 384 448  512 ... {A152691(n+3)}
  6: 448 576 704 832 960 1088 ... {64*A005408(n+3)}
  ...
The sequence s, the first (n=0) column of A, is always the binomial transform of the first (k=0) row in A.
A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j)+1) = 2^k*(k + 1 + 2*n), for k >= 0 and n >= 0.
The corresponding antidiagonal-upwards read triangle is T(k, n) = A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.
If the nonnegative integers A001477 are used as k = 0 row of the array Anneg(k, n) = 2^(k-1)*(2*n + k), for k >= 0, n >= 0, with the triangle Tnneg(k, n) = Anneg(k-n, n) = (n + k)*2^(k-n-1), k >= 0, n = 0..k, then the s sequence is snneg(k) = Tnneg(k, 0) = k*2^{k-1} = A001787(k), the binomial transform of the sequence{A001477(n)}_{n>=0}. The triangle Tnneg begins [0], [1, 1], [4, 3, 2], [12, 8, 5, 3], [32, 20, 12, 7, 4], ... . See A062111 and the row-reversed triangle A152920 for other versions.

			The triangle T(k, n) begins:
   k\n    0    1    2    3   4   5   6   7  8  9 10 ...
  -----------------------------------------------------
   0:     1
   1:     4    3
   2:    12    8    5
   3:    32   20   12    7
   4:    80   48   28   16   9
   5:   192  112   64   36  20  11
   6:   448  256  144   80  44  24  13
   7:  1024  576  320  176  96  52  28  15
   8:  2304 1280  704  384 208 112  60  32 17
   9:  5120 2816 1536  832 448 240 128  68 36 19
  10: 11264 6144 3328 1792 960 512 272 144 76 40 21
  ...

Column sequences without leading zeros are for n=0..9: A001787(n+1), A001792(n+1), A045623(n+2), A045891(n+3), A034007(n+4), A111297(n+3), A159694(n+1), A159695(n+1), A159696(n+1), A159697(n+1).
The sequence of (sub)diagonal k, for k >= 0, is the row k sequence of array A: {(k + 2*n + 1)*2^k}_{k >= 0}.
Row sums: A213569(k+1), k >= 0 (see the J. M. Bergot comments there).
Cf. A006211, A152920.

Table[2^#*(# + 1 + 2 n) &[k - n], {k, 0, 10}, {n, 0, k}] // Flatten (* Michael De Vlieger, Oct 03 2019 *)

Array A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j) + 1) = 2^k*(k + 1+ 2*n), for k >= 0 and n >= 0.
Triangle T(k, n) =  A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.
Recurrence: T(k, 0) = (k+1)*2^k = A001787(k+1), for k >= 0, and  T(k, n) = T(k, n-1) - T(k-1, n-1), for n >= 1, k >= 1, with T(k, n) = 0 if k < n.
O.g.f. for row polynomials: G(z,x) = Sum_{n=0..k} R(k, x)*z^n =
(1  + x*z*(1 - 4*z))/((1 - 2*z)^2*(1 - x*z)^2).
T(k, 0) = Sum_{n=0..k} binomial(k,n)*T(n, n), k >= 0 (binomial transform).

Definition corrected by Georg Fischer, Jul 13 2023

A374419 Triangle read by rows: T(n,k) = number of permutations in symmetric group S_n with an even number of non-fixed point cycles, without k<=n particular fixed points.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 4, 3, 3, 3, 3, 36, 32, 29, 26, 23, 20, 296, 260, 228, 199, 173, 150, 130, 2360, 2064, 1804, 1576, 1377, 1204, 1054, 924, 19776, 17416, 15352, 13548, 11972, 10595, 9391, 8337, 7413, 180544, 160768, 143352, 128000, 114452, 102480, 91885, 82494, 74157, 66744

Offset: 0

Julian Hatfield Iacoponi, Jul 08 2024

			Triangle array T(n,k) begins:
  n: {k<=n}
  0:  {1}
  1:  {1,       0}
  2:  {1,       0,       0}
  3:  {1,       0,       0,       0}
  4:  {4,       3,       3,       3,       3}
  5:  {36,      32,      29,      26,      23,      20}
  6:  {296,     260,     228,     199,     173,     150,     130}
  7:  {2360,    2064,    1804,    1576,    1377,    1204,    1054,   924}
T(n,0) = A373339(n) = the number of permutations in S_n without k=0 particular fixed points (i.e., not filtered, so all permutations) with an even number of cycles.
T(n,n) = A216778(n) = the number of permutations in S_n without k=n particular fixed points (i.e., filtered down to just the derangements) with an even number of cycles.
T(4,1<=k<=4) = 3 because S_4 contains 3 permutations with an even number of non-fixed point cycles without k=1,2,3 or 4 particular fixed points, namely the 3 (2,2)-cycles: (12)(34), (13)(24), (14)(23).
T(4,0) = 4 is one more than the above because it includes the permutation without k=0 particular fixed points, i.e., the identity permutation of 4 fixed points.

Cf. A047920, A062111.

Cf. A374420 (odd case), A216778 (main diagonal), A373339 (first column).

Table[Table[1/2*(Sum[(-1)^j*Binomial[k, j]*(n - j)!, {j, 0, k}] + 2^(n - k - 1)*(2 - n - k)), {k, 0, n}], {n, 0, 10}]

T(n,k) = T(n,k-1) - T(n-1,k-1) with T(n,0) = A373339(n).

T(n,k) = (1/2) * (Sum_{j=0..k} (-1)^j * binomial(k,j) * (n-j)! + 2^(n-k-1)*(2-n-k)).

A374420 Triangle T(n, k) for the number of permutations of symmetric group S_n with an odd number of non-fixed point cycles, without k <= n particular fixed points.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 5, 4, 3, 2, 20, 15, 11, 8, 6, 84, 64, 49, 38, 30, 24, 424, 340, 276, 227, 189, 159, 135, 2680, 2256, 1916, 1640, 1413, 1224, 1065, 930, 20544, 17864, 15608, 13692, 12052, 10639, 9415, 8350, 7420, 182336, 161792, 143928, 128320, 114628, 102576, 91937, 82522, 74172, 66752

Offset: 0

Julian Hatfield Iacoponi, Jul 08 2024

			Triangle array T(n,k)
n: {k<=n}
0:  {0}
1:  {0,       0}
2:  {1,       1,       1}
3:  {5,       4,       3,       2}
4:  {20,      15,      11,      8,       6}
5:  {84,      64,      49,      38,      30,      24}
6:  {424,     340,     276,     227,     189,     159,     135}
7:  {2680,    2256,    1916,    1640,    1413,    1224,    1065,   930}
T(n,0) = A373340(n) = the number of permutations in S_n without k=0 particular fixed points (i.e., not filtered, so all permutations) with an odd number of cycles.
T(n,n) = A216779(n) = the number of permutations in S_n without k=n particular fixed points (i.e., filtered down to just the derangements) with an odd number of cycles.
T(2,k) = 1 because S_2 contains 1 permutation with an odd number of non-fixed point cycles without k=0,1 or 2 particular fixed points, namely the derangement (12).
T(3,2) = 3 because S_3 contains 3 permutations with an odd number of non-fixed point cycles without k=2 particular fixed points: say, without fixed points (1) and (2), namely (12)(3), (123), (132).

Cf. A047920, A062111, A047920.

Cf. A374419 (even case), A216779 (main diagonal), A373340 (first column).

Table[Table[1/2*(Sum[(-1)^j*Binomial[k, j]*(n - j)!, {j, 0, k}] - 2^(n - k - 1)*(2 - n - k)), {k, 0, n}], {n, 0, 10}]

T(n,k) = T(n,k-1) - T(n-1,k-1) with T(n,0) = A373340(n).
T(n,k) = (1/2)*(Sum_{j=0..k} (-1)^j * binomial(k,j) * (n-j)! - 2^(n-k-1)*(2-n-k)).
T(n,k) = (A047920(n, k) + 2^(n-k-1)*(n+k-2))/2. - Peter Luschny, Jul 28 2024

cp's OEIS Frontend

A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers.

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A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k*(k + 1 + 2*n), k >= 0, n >= 0, read by antidiagonals upwards.

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A374419 Triangle read by rows: T(n,k) = number of permutations in symmetric group S_n with an even number of non-fixed point cycles, without k<=n particular fixed points.

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A374420 Triangle T(n, k) for the number of permutations of symmetric group S_n with an odd number of non-fixed point cycles, without k <= n particular fixed points.

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A159756 Triangle A159755 reversed .

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A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k(k + 1 + 2n), k >= 0, n >= 0, read by antidiagonals upwards.