A094727 -id:A094727 - cp's OEIS frontend

A214604 Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.

1, 5, 9, 11, 17, 25, 19, 27, 37, 49, 29, 39, 51, 65, 81, 41, 53, 67, 83, 101, 121, 55, 69, 85, 103, 123, 145, 169, 71, 87, 105, 125, 147, 171, 197, 225, 89, 107, 127, 149, 173, 199, 227, 257, 289, 109, 129, 151, 175, 201, 229, 259, 291, 325, 361, 131, 153, 177, 203, 231, 261, 293, 327, 363, 401, 441

Offset: 1

Reinhard Zumkeller, Jul 25 2012

			.     Take the first n elements of the n-th diagonal (northeast to
.     southwest) of the triangle on the left side
.     and write this as n-th row on the triangle of the right side.
. 1:                1                    1
. 2:              _   5                  5  9
. 3:            _   9  11               11 17 25
. 4:         __  __  17  19             19 27 37 49
. 5:       __  __  25  27  29           29 39 51 65 ..
. 6:     __  __  __  37  39  41         41 53 67 .. .. ..
. 7:   __  __  __  49  51  53  55       55 69 .. .. .. .. ..
. 8: __  __  __  __  65  67  69  71     71 .. .. .. .. .. .. .. .

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A000290, A016754, A028387, A025581, A094727, A176271.

Cf. A214659 (row sums), A214660 (main diagonal), A214661.

import Data.List (transpose)
a214604 n k = a214604_tabl !! (n-1) !! (k-1)
a214604_row n = a214604_tabl !! (n-1)
a214604_tabl = zipWith take [1..] $ transpose a176271_tabl

[(n+k)^2-n-3*k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

Table[(n+k)^2-n-3*k+1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[(n+k)^2-n-3*k+1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n,k) = (n+k)^2 - n - 3*k + 1.
Sum_{k=1..n} T(n, k) = A214659(n).
T(2*n-1, n) = A214660(n) (main diagonal).
T(n, 1) = A028387(n-1).
T(n, n) = A016754(n-1).
T(n, k) = A214661(n,k) + 2*A025581(n,k).
T(n, k) = 2*A000290(A094727(n,k)) - A214661(n,k).

A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

Original entry on oeis.org

1, 3, 9, 7, 15, 25, 13, 23, 35, 49, 21, 33, 47, 63, 81, 31, 45, 61, 79, 99, 121, 43, 59, 77, 97, 119, 143, 169, 57, 75, 95, 117, 141, 167, 195, 225, 73, 93, 115, 139, 165, 193, 223, 255, 289, 91, 113, 137, 163, 191, 221, 253, 287, 323, 361, 111, 135, 161, 189, 219, 251, 285, 321, 359, 399, 441

Offset: 1

Reinhard Zumkeller, Jul 25 2012

			.     Take the first n elements of the n-th diagonal (northwest to
.     southeast) of the triangle on the left side
.     and write this as n-th row on the triangle of the right side.
. 1:                1                    1
. 2:              3   _                  3  9
. 3:            7   9  __                7 15 25
. 4:         13  15  __  __             13 23 35 49
. 5:       21  23  25  __  __           21 33 47 63 ..
. 6:     31  33  35  __  __  __         31 45 61 .. .. ..
. 7:   43  45  47  49  __  __  __       43 59 .. .. .. .. ..
. 8: 57  59  61  63  __  __  __  __     57 .. .. .. .. .. .. .. .

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A000290, A002061, A016754, A025581, A094727, A176271, A214604.

Cf. A051673 (row sums), A214675 (main diagonal).

import Data.List (transpose)
a214661 n k = a214661_tabl !! (n-1) !! (k-1)
a214661_row n = a214661_tabl !! (n-1)
a214661_tabl = zipWith take [1..] $ transpose $ map reverse a176271_tabl

[(n+k)^2-3*n-k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

Table[(n+k)^2-3*n-k+1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[(n+k)^2-3*n-k+1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = (n+k)^2 - 3*n - k + 1.
T(n,k) = A176271(n+k-1, k).
T(n, k) = A214604(n,k) - 2*A025581(n,k).
T(n, k) = 2*A000290(A094727(n,k)) - A214604(n,k).
T(2*n-1, n) = A214675() (main diagonal).
T(n,1) = A002061(n).
T(n,n) = A016754(n-1).
Sum_{k=1..n} T(n, k) = A051673(n) (row sums).

A182630 T(n,k) = A002024(k+1)*n + A002262(k), n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 2, 2, 1, 3, 4, 3, 0, 4, 6, 5, 3, 1, 5, 8, 7, 6, 4, 2, 6, 10, 9, 9, 7, 5, 0, 7, 12, 11, 12, 10, 8, 4, 1, 8, 14, 13, 15, 13, 11, 8, 5, 2, 9, 16, 15, 18, 16, 14, 12, 9, 6, 3, 10, 18, 17, 21, 19, 17, 16, 13, 10, 7, 0

Offset: 0

Omar E. Pol, Dec 06 2010

A table of congruences.

See A182631 for another version.

			Table of congruences:
===============+====+=======+==========+=============+====
           mod |  1 |   2   |     3    |      4      | ...
===============+====+=======+==========+=============+====
  congruent to |  0 |  0  1 |  0  1  2 |  0  1  2  3 | ...
===============+====+=======+==========+=============+====
Array begins:  |  0 |  0  1 |  0  1  2 |  0  1  2  3 | ...
               |  1 |  2  3 |  3  4  5 |  4  5  6  7 | ...
               |  2 |  4  5 |  6  7  8 |  8  9 10 11 | ...
               |  3 |  6  7 |  9 10 11 | 12 13 14 15 | ...
               |  4 |  8  9 | 12 13 14 | 16 17 18 19 | ...
               |  5 | 10 11 | 15 16 17 | 20 21 22 23 | ...
               |  6 | 12 13 | 18 19 20 | 24 25 26 27 | ...
               |  7 | 14 15 | 21 22 23 | 28 29 30 31 | ...
               |  8 | 16 17 | 24 25 26 | 32 33 34 35 | ...
               |  9 | 18 19 | 27 28 29 | 36 37 38 39 | ...
               | 10 | 20 21 | 30 31 32 | 40 41 42 43 | ...

Columns: A001477, A005483, A005408, A008585, A016777, A016789, A008586, A016813, A016825, A004767.
Rows: A002262, A094727, A182815.
Cf. A182631.

A002262 := proc(n) n-binomial(floor(1/2+sqrt(2+2*n)),2) ; end proc:
A002024 := proc(n) ceil((sqrt(1+8*n)-1)/2) ;end proc:
A182630 := proc(n,k) A002024(k+1)*n+A002262(k) ; end proc: # R. J. Mathar, Dec 09 2010

A193657 First difference of A002627.

Original entry on oeis.org

1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Previous name was: Q-residue of the triangle A094727, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
Number of n X n rook placements avoiding the pattern 001. - N. J. A. Sloane, Feb 04 2013
Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 2, 3, etc., along the main diagonal, and zeros everywhere else. Then a(n) is equal to the permanent of M(n). - John M. Campbell, Apr 20 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013

Cf. A193649, A094727, A082425, A114870, A176305, A296964.

a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1,1):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 30 2014

q[n_, k_] := n + k + 1;  (* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193657 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
CoefficientList[Series[(E^x-x)/(x-1)^2,{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Nov 20 2012 *)

a(n) = { sum(k=0, n, if (k <= n-2, binomial(n,k)*(k+1)!, binomial(n,k)^2*k!));} \\ Michel Marcus, Feb 07 2013

def A193657():
    a = 2; b = 7; c = 31; n = 3
    yield 1
    while True:
        yield a
        n += 1
        a,b,c = b,c,((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
a = A193657(); [next(a) for n in range(19)] # Peter Luschny, May 30 2014

E.g.f.: (exp(x)-x)/(x-1)^2. - Vaclav Kotesovec, Nov 20 2012
a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012
a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - Peter Luschny, May 30 2014
a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 30 2014
From Peter Bala, Feb 10 2020: (Start)
a(n) = n*A002627(n) + 1.
a(n) = A114870(n) + n!.
a(n) = A296964(n+1) - A296964(n) for n >= 2.
a(1) = 2 and a(n) = (n^2*a(n-1) - 1)/(n - 1) for n >= 2. See A082425 for solutions to this recurrence with different starting values.
Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
Second column of A176305. (End)

Simpler definition by Peter Luschny, May 30 2014

A319556 a(n) gives the alternating sum of length n, starting at n: n - (n+1) + (n+2) - ... + (-1)^(n+1) * (2n-1).

Original entry on oeis.org

1, -1, 4, -2, 7, -3, 10, -4, 13, -5, 16, -6, 19, -7, 22, -8, 25, -9, 28, -10, 31, -11, 34, -12, 37, -13, 40, -14, 43, -15, 46, -16, 49, -17, 52, -18, 55, -19, 58, -20, 61, -21, 64, -22, 67, -23, 70, -24, 73, -25, 76, -26, 79, -27, 82, -28, 85, -29, 88, -30

Offset: 1

Mark Povich, Aug 27 2019

As can be observed from Bernard Schott's formula, and also proved using elementary methods of slope and angle determination, extending the graph of this sequence forms two lines (given by y = 1.5x - 0.5 and y = -0.5x) that intersect at (0.25, -0.125) in an angle of intersection of ~82.87 degrees. The angles of incidence of these lines off the horizontal axis are ~56.31 and ~-26.56 degrees.

If one wished to include negative input values, one could proceed, e.g., -3+4-5 (=-8) or -3+2-1 (=-2). If the former, then the sequence merely switches signs for negative inputs, graphically extending the previous lines to the left of the vertical. If the latter, two new lines emerge left of the vertical, both of slope 1/2. Increasing the run in this case "spreads apart" all y-intercepts.

			If n=5, a(n)=7, since 5-6+7-8+9 = 7.
If n=6, a(n)=-3, since 6-7+8-9+10-11 = -3.

Mark Povich, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A094727, A123684, A128622.

[((2*n-1)*(n mod 2) - n*(-1)^n)/2: n in [0..70]]; // G. C. Greubel, Mar 14 2024

LinearRecurrence[{0,2,0,-1}, {1,-1,4,-2}, 60] (* Metin Sariyar, Sep 15 2019 *)

a(n) = sum(k=n, 2*n-1, (-1)^(n-k)*k); \\ Michel Marcus, Aug 27 2019

Vec(x*(1 - x + 2*x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60)) \\ Colin Barker, Sep 07 2019

def alt(k):
    return sum(k[::2])-sum(k[1::2])
def alt_run(n):
    m = []
    m.append(n)
    for i in range (1, n):
        m.append(m[0]+i)
    return alt(m)
t=[]
for i in range (100):
    t.append(alt_run(i))
print(t)

[((2*n-1)*(n%2) - n*(-1)^n)/2 for n in range(1,71)] # G. C. Greubel, Mar 14 2024

From Bernard Schott, Aug 27 2019: (Start)
a(2*n-1) = 3*n-2 for n >= 1,
a(2*n) = - n for n >= 1. (End)
a(n) = Sum_{k=n..2*n-1} (-1)^(n-k)*k.
From Colin Barker, Sep 07 2019: (Start)
G.f.: x*(1 - x + 2*x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>4.
a(n) = ((2*n-1)*(1 - (-1)^n) - 2*n*(-1)^n)/4. (End)
E.g.f.: (1/4)*((1 + 4*x)*exp(-x) - (1 - 2*x)*exp(x)). - Stefano Spezia, Sep 07 2019 after Colin Barker
From G. C. Greubel, Mar 14 2024: (Start)
a(n) = Sum_{k=0..n-1} (-1)^k*A094727(n, k).
a(n) = Sum_{k=1..n} (-1)^(k-1)*A128622(n, k). (End)

A334016 Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess queen from (1,1) to (n,k) in the first quadrant using only right, diagonal up-right, and diagonal up-left moves.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 4, 10, 21, 35, 8, 25, 65, 139, 237, 16, 60, 179, 451, 978, 1684, 32, 140, 470, 1337, 3339, 7239, 12557, 64, 320, 1189, 3725, 10325, 25559, 55423, 96605, 128, 720, 2926, 9958, 30018, 81716, 200922, 435550, 761938, 256, 1600, 7048, 25802, 83518

Offset: 1

Peter Kagey, Apr 12 2020

			Table begins:
n\k|   1    2     3      4       5        6         7          8
---+------------------------------------------------------------
  1|   1    1     6     35     237     1684     12557      96605
  2|   1    4    21    139     978     7239     55423     435550
  3|   2   10    65    451    3339    25559    200922    1611624
  4|   4   25   179   1337   10325    81716    658918    5394051
  5|   8   60   470   3725   30018   245220   2027447   16935981
  6|  16  140  1189   9958   83518   703635   5961973   50811786
  7|  32  320  2926  25802  224831  1951587  16938814  147261146
  8|  64  720  7048  65241  589701  5269220  46826316  415175289
For example, the T(2,2) = 4 valid sequences of moves from (1,1) to (2,2) are:
(1,1) -> (2,1) -> (1,2) -> (2,2),
(1,1) -> (2,1) -> (3,1) -> (2,2),
(1,1) -> (2,2), and
(1,1) -> (3,1) -> (2,2).

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals)
Peter Kagey, Parity bitmap of first 2048 rows and 1024 columns. (Even and odd entries and represented by black and white pixels respectively.)

Cf. A035002 (up, right), A059450 (right, up-left), A132439 (up, right, up-right), A279212 (up, right, up-right, up-left), A334017 (up, right, up-left).

A071945 is the analog for king moves. For both king and queen moves, A094727 is the length of the longest sequence of moves.

T(n,k) = Sum_{i=1..k-1} T(n+i, k-i) + Sum_{i=1..min(n,k)-1} T(n-i, k-i) + Sum_{i=1..n-1} T(n-i, k).

A334017 Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess queen from (1,1) to (n,k) in the first quadrant using only up, right, and diagonal up-left moves.

Original entry on oeis.org

1, 1, 2, 2, 5, 10, 4, 13, 33, 63, 8, 32, 98, 240, 454, 16, 76, 269, 777, 1871, 3539, 32, 176, 702, 2295, 6420, 15314, 29008, 64, 400, 1768, 6393, 19970, 54758, 129825, 246255, 128, 896, 4336, 17088, 58342, 176971, 478662, 1129967, 2145722, 256, 1984, 10416

Offset: 1

Peter Kagey, Apr 12 2020

First row is A175962.

			Table begins:
n\k|  1   2     3      4       5        6         7          8
---+----------------------------------------------------------
  1|  1   2    10     63     454     3539     29008     246255
  2|  1   5    33    240    1871    15314    129825    1129967
  3|  2  13    98    777    6420    54758    478662    4266102
  4|  4  32   269   2295   19970   176971   1593093   14532881
  5|  8  76   702   6393   58342   536080   4965056   46345046
  6| 16 176  1768  17088  163041  1550809  14765863  140982374
  7| 32 400  4336  44280  440602  4332221  42373370  413689403
  8| 64 896 10416 111984 1159580 11771312 118190333 1179448443
For example, the T(2,2) = 5 sequences of permissible queen's moves from (1,1) to (2,2) are:
(1,1) -> (1,2) -> (2,2),
(1,1) -> (2,1) -> (1,2) -> (2,2),
(1,1) -> (2,1) -> (2,2),
(1,1) -> (2,1) -> (3,1) -> (2,2), and
(1,1) -> (3,1) -> (2,2).

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals)
Peter Kagey, Parity bitmap for first 1024 rows and columns. (Even and odd entries and represented by black and white pixels respectively.)

Cf. A175962.
Cf. A035002 (up, right), A059450 (right, up-left), A132439 (up, right, up-right), A279212 (up, right, up-left), A334016 (right, up-right, up-left).
A033877 is the analog for king moves. For both king and queen moves, A094727 is the length of the longest sequence of moves.

A102105 a(n) = (195^n - 163^n + 1) / 4.

Original entry on oeis.org

1, 12, 83, 486, 2645, 13872, 71303, 362346, 1829225, 9198612, 46150523, 231225006, 1157542205, 5791962552, 28972567343, 144901100466, 724620293585, 3623445841692, 18118262329763, 90594411012726, 452981353155365, 2264934660052032, 11324756983085783

Offset: 0

Gary W. Adamson, Dec 30 2004

Sum of the entries in the last row of the 3 X 3 matrix M^n, where M = {{1, 0, 0}, {2, 3, 0}, {3, 4, 5}}.

Sum of the entries in the second row of M^n = A048473(n).

			a(4) = 2645 = 9*486 - 23*83 + 15*12 = 9*a(3) - 23*a(2) + 15*a(1).
a(4) = 2645 since M^4 * {1, 1, 1} = {1, 161, 2645}, where 161 = A048473(4).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A000326, A048473, A094727.

List([0..30], n-> (19*5^n -16*3^n +1)/4); # G. C. Greubel, Oct 27 2019

[(19*5^n -16*3^n +1)/4: n in [0..30]]; // G. C. Greubel, Oct 27 2019

with(linalg): M[1]:=matrix(3,3,[1,0,0,2,3,0,3,4,5]): for n from 2 to 23 do M[n]:=multiply(M[1],M[n-1]) od: 1,seq(multiply(M[n],matrix(3,1,[1,1,1]))[3,1],n=1..23);
seq((19*5^n -16*3^n +1)/4, n=0..30); # G. C. Greubel, Oct 27 2019

Table[(19*5^n -16*3^n +1)/4, {n,0,30}] (* G. C. Greubel, Oct 27 2019 *)
LinearRecurrence[{9,-23,15},{1,12,83},30] (* Harvey P. Dale, Sep 19 2021 *)

Vec((1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)) + O(x^30)) \\ Colin Barker, Mar 03 2017

[(19*5^n -16*3^n +1)/4 for n in (0..30)] # G. C. Greubel, Oct 27 2019

a(n) = 9*a(n-1) - 23*a(n-2) + 15*a(n-3), a(0)=1,a(1)=12,a(2)=83 (derived from the minimal polynomial of the matrix M).
G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)). - Colin Barker, Mar 03 2017
E.g.f.: (exp(x) - 16*exp(3*x) + 19*exp(5*x))/4. - G. C. Greubel, Oct 27 2019

Corrected by T. D. Noe, Nov 07 2006
Edited by N. J. A. Sloane, Dec 02 2006
New definition from Ralf Stephan, May 17 2007

A182815 The third row of table A182630.

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 8, 9, 10, 11, 10, 11, 12, 13, 14, 12, 13, 14, 15, 16, 17, 14, 15, 16, 17, 18, 19, 20, 16, 17, 18, 19, 20, 21, 22, 23, 18, 19, 20, 21, 22, 23, 24, 25, 26, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 26

Offset: 0

Omar E. Pol, Dec 06 2010

Cf. A002262, A094727, A182630, A182631.

Maple
```
seq(A182630(2,n),n=0..80) ;
```

a(n) = A182630(2,n).

A347823 Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 3, 8, 5, 4, 15, 18, 7, 5, 24, 42, 32, 9, 6, 35, 80, 90, 50, 11, 7, 48, 135, 200, 165, 72, 13, 8, 63, 210, 385, 420, 273, 98, 15, 9, 80, 308, 672, 910, 784, 420, 128, 17, 10, 99, 432, 1092, 1764, 1890, 1344, 612, 162, 19, 11, 120, 585, 1680, 3150, 4032, 3570, 2160, 855, 200, 21

Offset: 0

Jules Beauchamp, Jan 23 2022

			Triangle begins:
  1;
  2,  3;
  3,  8,   5;
  4, 15,  18,   7;
  5, 24,  42,  32,   9;
  6, 35,  80,  90,  50,  11;
  7, 48, 135, 200, 165,  72, 13;
  8, 63, 210, 385, 420, 273, 98, 15;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums give A053220.
Columns give A000027, A005563, A212343.
Diagonals give A005408, A001105, A059270, A112742.
Cf. A007318, A094727.

T(n,k) = (n+k+1)*binomial(n,k) \\ Andrew Howroyd, Jan 23 2022

T(n,k) = A094727(n+1,k)*A007318(n,k).

Row g.f.: (1 + x)^(n-1)*(1 + n + x + 2*n*x). - Stefano Spezia, Jan 23 2022

cp's OEIS Frontend

A214604 Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.

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A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

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A182630 T(n,k) = A002024(k+1)*n + A002262(k), n >= 0, k >= 0, read by antidiagonals.

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Maple

A193657 First difference of A002627.

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A319556 a(n) gives the alternating sum of length n, starting at n: n - (n+1) + (n+2) - ... + (-1)^(n+1) * (2n-1).

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A334016 Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess queen from (1,1) to (n,k) in the first quadrant using only right, diagonal up-right, and diagonal up-left moves.

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A334017 Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess queen from (1,1) to (n,k) in the first quadrant using only up, right, and diagonal up-left moves.

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A102105 a(n) = (195^n - 163^n + 1) / 4.