A006012 -id:A006012 - cp's OEIS frontend

A191348 Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 2, 1, 0, 8, 20, 7, 2, 1, 0, 16, 68, 26, 8, 3, 1, 0, 32, 232, 97, 32, 14, 3, 1, 0, 64, 792, 362, 128, 72, 15, 3, 1, 0, 128, 2704, 1351, 512, 376, 81, 16, 3, 1, 0

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,     0,      0,      0,       0,        0,         0, ...
1, 1,  2,   4,    8,    16,     32,     64,     128,      256,       512, ...
1, 2,  6,  20,   68,   232,    792,   2704,    9232,    31520,    107616, ...
1, 2,  7,  26,   97,   362,   1351,   5042,   18817,    70226,    262087, ...
1, 2,  8,  32,  128,   512,   2048,   8192,   32768,   131072,    524288, ...
1, 3, 14,  72,  376,  1968,  10304,  53952,  282496,  1479168,   7745024, ...
1, 3, 15,  81,  441,  2403,  13095,  71361,  388881,  2119203,  11548575, ...
1, 3, 16,  90,  508,  2868,  16192,  91416,  516112,  2913840,  16450816, ...
1, 3, 17,  99,  577,  3363,  19601, 114243,  665857,  3880899,  22619537, ...
1, 3, 18, 108,  648,  3888,  23328, 139968,  839808,  5038848,  30233088, ...
1, 4, 26, 184, 1316,  9424,  67496, 483424, 3462416, 24798784, 177615776, ...
1, 4, 27, 196, 1433, 10484,  76707, 561236, 4106353, 30044644, 219825387, ...
1, 4, 28, 208, 1552, 11584,  86464, 645376, 4817152, 35955712, 268377088, ...
1, 4, 29, 220, 1673, 12724,  96773, 736012, 5597777, 42574180, 323800109, ...
1, 4, 30, 232, 1796, 13904, 107640, 833312, 6451216, 49943104, 386642400, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A006012, row 4 is A001075, row 5 is A081294, row 6 is A098648, row 7 is A084120, row 8 is A146963, row 9 is A001541, row 10 is A081341, row 11 is A084134, row 13 is A090965.
Row 3*2 is A056236, row 4*2 is A003500, row 5*2 is A155543, row 9*2 is A003499.
Cf. A191347 which uses floor() in place of ceiling().

T(n, k) = if (k==0, 1, if (k==1, ceil(sqrt(n)), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 23 2019

For each row n >= 0 let T(n,0)=1 and T(n,1) = ceiling(sqrt(n)), then for each column k >= 2: T(n,k) = T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 23 2019

A082388 a(1) = 1, a(2) = 2; further terms are defined by rules that for k >= 2, a(2^k-i) = a(2^k+i) for 1 <= i <= 2^k-1 and a(2^k) = a(2^(k-1)) + Sum_{i < 2^k} a(i).

Original entry on oeis.org

1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 68, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 232, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 68, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 792, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 68, 1, 2, 1, 6, 1, 2, 1, 20, 1, 2, 1, 6, 1, 2, 1, 232, 1, 2, 1

Offset: 1

Benoit Cloitre, Apr 14 2003

Andrew Howroyd, Table of n, a(n) for n = 1..1024

Cf. A006012.

a[n_] := With[{e = IntegerExponent[n, 2]}, Sum[Binomial[e, 2k] 2^(e-k), {k, 0, Quotient[e, 2]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)

a(n)={my(e=valuation(n,2)); sum(k=0, e\2, binomial(e, 2*k)*2^(e-k))} \\ Andrew Howroyd, Jul 31 2018

a(2^k) = 4*a(2^(k-1)) - 2*a(2^(k-2));
a(2^k) = round((1/2)*(2+sqrt(2))^k).
Multiplicative with a(2^e) = A006012(e), a(p^e) = 1 for odd prime p. - Andrew Howroyd, Jul 31 2018

A124216 Generalized Pascal triangle.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 34, 16, 1, 1, 25, 90, 90, 25, 1, 1, 36, 195, 328, 195, 36, 1, 1, 49, 371, 931, 931, 371, 49, 1, 1, 64, 644, 2240, 3334, 2240, 644, 64, 1, 1, 81, 1044, 4788, 9846, 9846

Offset: 0

Paul Barry, Oct 19 2006

Consider the 1-parameter family of triangles with g.f. (1-x(1+y))/(1-2x(1+y)+x^2(1+k*x+y^2)). A007318 corresponds to k=2. A056241 corresponds to k=1. A124216 corresponds to k=0. Row sums are A006012. Diagonal sums are A124217.

			Triangle begins
1,
1, 1,
1, 4, 1,
1, 9, 9, 1,
1, 16, 34, 16, 1,
1, 25, 90, 90, 25, 1,
1, 36, 195, 328, 195, 36, 1,
1, 49, 371, 931, 931, 371, 49, 1

P. Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5

Cf. A001263.

G.f.: (1-x(1+y))/(1-2x(1+y)+x^2(1+y^2)); Number triangle T(n,k)=sum{j=0..n, C(n,j)C(j,2(j-k))2^(j-k)}.

Equals 2*A001263 - A007318; (i.e. twice the Narayana triangle minus Pascal's triangle). - Gary W. Adamson, Jun 14 2007

A152599 a(n) = 10a(n-1) - 12a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

Original entry on oeis.org

1, 4, 28, 232, 1984, 17056, 146752, 1262848, 10867456, 93520384, 804794368, 6925699072, 59599458304, 512886194176, 4413668442112, 37982050091008, 326856479604736, 2812780194955264, 24205524194295808, 208301879603494912, 1792552505703399424, 15425902501792055296

Offset: 0

Philippe Deléham, Dec 09 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-12).

Cf. A052961, A006012, A147703, A152596.

LinearRecurrence[{10, -12}, {1, 4}, 25] (* Paolo Xausa, Jan 19 2024 *)

G.f.: (1-6*x)/(1-10*x+12*x^2).
a(n) = Sum_{k=0..n} A147703(n,k)*3^(n-k).
a(n) = 2^n*A052961(n). - R. J. Mathar, Jun 14 2016

A221337 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some king-move neighbor, with every occupancy equal to zero or two.

Original entry on oeis.org

0, 2, 2, 0, 6, 0, 4, 20, 20, 4, 0, 68, 0, 68, 0, 8, 232, 790, 790, 232, 8, 0, 792, 0, 10704, 0, 792, 0, 16, 2704, 34042, 142792, 142792, 34042, 2704, 16, 0, 9232, 0, 1937900, 0, 1937900, 0, 9232, 0, 32, 31520, 1470618, 26264018

Offset: 1

R. H. Hardin Jan 11 2013

Table starts
..0....2.....0.......4......0.......8........0......16.....0.32
..2....6....20......68....232.....792.....2704....9232.31520
..0...20.....0.....790......0...34042........0.1470618
..4...68...790...10704.142792.1937900.26264018
..0..232.....0..142792......0
..8..792.34042.1937900
..0.2704.....0
.16.9232
..0

			Some solutions for n=3 k=4
..0..2..2..2....2..0..0..0....2..0..2..0....0..0..2..0....0..2..2..0
..2..0..0..0....2..2..2..0....0..0..2..0....2..2..0..0....2..2..0..0
..0..2..0..2....0..2..0..2....2..0..2..2....2..0..2..2....0..0..2..2

Column 1 is A077957

Column 2 is A006012

A361432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 20, 8, 0, 1, 5, 20, 54, 68, 16, 0, 1, 6, 30, 112, 252, 232, 32, 0, 1, 7, 42, 200, 656, 1188, 792, 64, 0, 1, 8, 56, 324, 1400, 3904, 5616, 2704, 128, 0, 1, 9, 72, 490, 2628, 10000, 23360, 26568, 9232, 256, 0

Offset: 0

Seiichi Manyama, Mar 11 2023

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  1,   2,    3,    4,     5, ...
  0,  2,   6,   12,   20,    30, ...
  0,  4,  20,   54,  112,   200, ...
  0,  8,  68,  252,  656,  1400, ...
  0, 16, 232, 1188, 3904, 10000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Column k=0..11 give A000007, A011782, A006012, A083881, A081335, A090139, A145301, A145302, A145303, A143079, A289414, A289415.
Main diagonal gives A084062.
Cf. A084061, A084097.

T(n,k) = sum(j=0, n\2, k^(n-j)*binomial(n, 2*j));

T(n, k) = round(((k+sqrt(k))^n+(k-sqrt(k))^n))/2;

T(0,k) = 1, T(1,k) = k; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n + (k - sqrt(k))^n)/2.
G.f. of column k: (1 - k * x)/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * cosh(sqrt(k) * x).

A052680 Expansion of e.g.f. (1-2x)/(1-4x+2*x^2).

Original entry on oeis.org

1, 2, 12, 120, 1632, 27840, 570240, 13628160, 372234240, 11437977600, 390516940800, 14666390323200, 600890263142400, 26670379902566400, 1274817218759884800, 65287473566515200000, 3566486043252228096000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 628

Cf. A000129, A002203, A006012.

[Factorial(n)*(&+[Binomial(n, 2*k)*2^(n-k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 10 2022

spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Union(Z,Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-2x)/(1-4x+2x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 28 2019 *)
Table[n!*2^(n/2)*ChebyshevT[n, Sqrt[2]], {n,0,50}] (* G. C. Greubel, Jun 10 2022 *)

[2^(n/2)*factorial(n)*chebyshev_T(n, sqrt(2)) for n in (0..50)] # G. C. Greubel, Jun 10 2022

E.g.f.: (1 - 2*x)/(1 - 4*x + 2*x^2).
D-finite with Recurrence: a(0)=1, a(1)=2, a(n+2) = 4*(n+2)*a(n+1) - 2*(2 +3*n +n^2)*a(n).
a(n) = (n!/2)*Sum_{alpha=RootOf(1 - 4*Z + 2*Z^2)} alpha^(-n).
a(n) = n!*A006012(n). - R. J. Mathar, Nov 27 2011
From G. C. Greubel, Jun 10 2022: (Start)
a(2*n) = (2*n)! * 2^(n-1)*A002203(2*n).
a(2*n+1) = (2*n+1)! * 2^(n+1)*A000129(2*n+1).
a(n) = 2^(n/2) * n! * ChebyshevT(n, sqrt(2)). (End)

A084867 Symmetric square table, read by antidiagonals, such that antidiagonal sums form the first row shifted left: T(0,0)=1, T(0,k) = Sum_{m=0..k-1} T(m,k-1-m) when k > 0; and T(n,k) = T(n-1,k) + T(n,k-1) when n > 0, k > 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 6, 4, 4, 6, 20, 10, 8, 10, 20, 68, 30, 18, 18, 30, 68, 232, 98, 48, 36, 48, 98, 232, 792, 330, 146, 84, 84, 146, 330, 792, 2704, 1122, 476, 230, 168, 230, 476, 1122, 2704, 9232, 3826, 1598, 706, 398, 398, 706, 1598, 3826, 9232, 31520, 13058, 5424

Offset: 0

Paul D. Hanna, Jun 10 2003, Jun 11 2003

Antidiagonal sums give A006012. Table is symmetric under transpose, so that first column equals the first row. Second row gives partial sums of first row.

			Table begins:
     1,     1,     2,     6,    20,    68,   232,    792, ...
     1,     2,     4,    10,    30,    98,   330,   1122, ...
     2,     4,     8,    18,    48,   146,   476,   1598, ...
     6,    10,    18,    36,    84,   230,   706,   2304, ...
    20,    30,    48,    84,   168,   398,  1104,   3408, ...
    68,    98,   146,   230,   398,   796,  1900,   5308, ...
   232,   330,   476,   706,  1104,  1900,  3800,   9108, ...
   792,  1122,  1598,  2304,  3408,  5308,  9108,  18216, ...
  2704,  3826,  5424,  7728, 11136, 16444, 25552,  43768, ...
  9232, 13058, 18482, 26210, 37346, 53790, 79342, 123110, ...

Cf. A006012 (row sums), A084868 (main diagonal).

T(0,0)=1, T(0,1)=1, T(0,n) = 4*T(0,n-1) - 2*T(0,n-2) when n >= 2.

A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 7, 20, 27, 13, 1, 9, 35, 73, 80, 34, 1, 11, 54, 151, 252, 234, 89, 1, 13, 77, 269, 597, 837, 677, 233, 1, 15, 104, 435, 1199, 2225, 2702, 1941, 610, 1, 17, 135, 657, 2158, 4956, 7943, 8533, 5523, 1597

Offset: 0

Philippe Deléham, Nov 06 2011

Mirror image of triangle in A147703.

			Triangle begins:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  9,  5;
  1,  7, 20, 27, 13;
  1,  9, 35, 73, 80, 34;

Cf. A001519, A059502, A000012, A005408, A014107.

Sum_{k=0..n} T(n,k)*x^k = A152620(n), A152594(n), A000007(n), A000012(n), A006012(n), A152596(n), A152599(n) for x=-3,-2,-1,0,1,2,3 respectively.
T(n,n) = A001519(n).
G.f.: (1-2y*x)/(1-(1+3y)*x+y*(1+y)*x^2).

A331969 T(n, k) = [x^(n-k)] 1/(((1 - 2x)^k)(1 - x)^(k + 1)). Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 7, 1, 1, 26, 30, 10, 1, 1, 57, 102, 58, 13, 1, 1, 120, 303, 256, 95, 16, 1, 1, 247, 825, 955, 515, 141, 19, 1, 1, 502, 2116, 3178, 2310, 906, 196, 22, 1, 1, 1013, 5200, 9740, 9078, 4746, 1456, 260, 25, 1

Offset: 0

Peter Luschny, Feb 03 2020

The triangle is the matrix inverse of the Riordan square (see A321620) generated by (1 + x - sqrt(1 - 6*x + x^2))/(4*x) (see A172094), where we take the absolute value of the terms.

T(n,k) is the number of evil-avoiding (2413, 3214, 4132, and 4213 avoiding) permutations of length (n+2) that start with 1 and whose inverse has k descents. - Donghyun Kim, Aug 16 2021

			Triangle starts:
[0] [1]
[1] [1,    1]
[2] [1,    4,    1]
[3] [1,   11,    7,    1]
[4] [1,   26,   30,   10,    1]
[5] [1,   57,  102,   58,   13,    1]
[6] [1,  120,  303,  256,   95,   16,    1]
[7] [1,  247,  825,  955,  515,  141,   19,   1]
[8] [1,  502, 2116, 3178, 2310,  906,  196,  22,  1]
[9] [1, 1013, 5200, 9740, 9078, 4746, 1456, 260, 25, 1]
...
Seen as a square array (the triangle is formed by descending antidiagonals):
1,  1,   1,    1,    1,     1,      1,      1,       1, ... [A000012]
1,  4,  11,   26,   57,   120,    247,    502,    1013, ... [A000295]
1,  7,  30,  102,  303,   825,   2116,   5200,   12381, ... [A045889]
1, 10,  58,  256,  955,  3178,   9740,  28064,   77093, ... [A055583]
1, 13,  95,  515, 2310,  9078,  32354, 106970,  333295, ...
1, 16, 141,  906, 4746, 21504,  87374, 326084, 1136799, ...
1, 19, 196, 1456, 8722, 44758, 204204, 849180, 3275931, ...

Donghyun Kim and Lauren Williams, Schubert polynomials and the inhomogeneous TASEP on a ring, arXiv:2102.00560 [math.CO], 2021.

Row sums A006012, alternating row sums A118434 with different signs, central column A091527.
T(n, 1) = A000295(n+1) for n >= 1, T(n, 2) = A045889(n-2) for n >= 2, T(n, 3) = A055583(n-3) for n >= 3.
Cf. A172094 (inverse up to sign).

gf := k -> 1/(((1-2*x)^k)*(1-x)^(k+1)): ser := k -> series(gf(k), x, 32):
# Prints the triangle:
seq(lprint(seq(coeff(ser(k), x, n-k), k=0..n)), n=0..6);
# Prints the square array:
seq(lprint(seq(coeff(ser(k), x, n), n=0..8)), k=0..6);

(* The function RiordanSquare is defined in A321620; returns the triangle as a lower triangular matrix. *)
M := RiordanSquare[(1 + x - Sqrt[1 - 6 x + x^2])/(4 x), 9];
Abs[#] & /@ Inverse[PadRight[M]]

cp's OEIS Frontend

A191348 Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

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A082388 a(1) = 1, a(2) = 2; further terms are defined by rules that for k >= 2, a(2^k-i) = a(2^k+i) for 1 <= i <= 2^k-1 and a(2^k) = a(2^(k-1)) + Sum_{i < 2^k} a(i).

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A124216 Generalized Pascal triangle.

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A152599 a(n) = 10*a(n-1) - 12*a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

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A221337 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some king-move neighbor, with every occupancy equal to zero or two.

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A361432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

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A052680 Expansion of e.g.f. (1-2*x)/(1-4*x+2*x^2).

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A084867 Symmetric square table, read by antidiagonals, such that antidiagonal sums form the first row shifted left: T(0,0)=1, T(0,k) = Sum_{m=0..k-1} T(m,k-1-m) when k > 0; and T(n,k) = T(n-1,k) + T(n,k-1) when n > 0, k > 0.

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A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A152599 a(n) = 10a(n-1) - 12a(n-2) for n > 1; a(0) = 1, a(1) = 4 .

A052680 Expansion of e.g.f. (1-2x)/(1-4x+2*x^2).

A331969 T(n, k) = [x^(n-k)] 1/(((1 - 2x)^k)(1 - x)^(k + 1)). Triangle read by rows, for 0 <= k <= n.