A190510 -id:A190510 - cp's OEIS frontend

A099089 Riordan array (1, 2+x).

1, 0, 2, 0, 1, 4, 0, 0, 4, 8, 0, 0, 1, 12, 16, 0, 0, 0, 6, 32, 32, 0, 0, 0, 1, 24, 80, 64, 0, 0, 0, 0, 8, 80, 192, 128, 0, 0, 0, 0, 1, 40, 240, 448, 256, 0, 0, 0, 0, 0, 10, 160, 672, 1024, 512, 0, 0, 0, 0, 0, 1, 60, 560, 1792, 2304, 1024, 0, 0, 0, 0, 0, 0, 12, 280, 1792, 4608, 5120, 2048

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A000129. Diagonal sums are A008346. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/2, -1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008
As an upper right triangle (in the example), table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber, May 08 2009

			Triangle begins:
  1;
  0,  2;
  0,  1,  4;
  0,  0,  4,  8;
  0,  0,  1, 12, 16;
  0,  0,  0,  6, 32, 32;
  0,  0,  0,  1, 24, 80, 64;
The entries can also be interpreted as the antidiagonal reading of the following array:
  1,    2,    4,    8,   16,   32,   64,  128,  256,  512, 1024,... A000079
  0,    1,    4,   12,   32,   80,  192,  448, 1024, 2304, 5120,... A001787
  0,    0,    1,    6,   24,   80,  240,  672, 1792, 4608,11520,... A001788
  0,    0,    0,    1,    8,   40,  160,  560, 1792, 5376,15360,... A001789
  0,    0,    0,    0,    1,   10,   60,  280, 1120, 4032,13440,...
  0,    0,    0,    0,    0,    1,   12,   84,  448, 2016, 8064,...
  0,    0,    0,    0,    0,    0,    1,   14,  112,  672, 3360,...
  0,    0,    0,    0,    0,    0,    0,    1,   16,  144,  960,...
  0,    0,    0,    0,    0,    0,    0,    0,    1,   18,  180,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    1,   20,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    0,    1,...

H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.

Eric W. Weisstein's Mathworld, Hypercube.

Cf. A053118, A008312, A062715, A038207.

Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.
G.f.: 1/(1-2y*x-y*x^2). - Philippe Deléham, Nov 20 2011
Sum_ {k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A090017(n+1), A090018(n), A190510(n+1), A190955(n+1) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 20 2011
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = 1, T(2,2) = 4, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A342134 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx - k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 18, 12, 0, 1, 8, 39, 80, 29, 0, 1, 10, 68, 252, 356, 70, 0, 1, 12, 105, 576, 1629, 1584, 169, 0, 1, 14, 150, 1100, 4880, 10530, 7048, 408, 0, 1, 16, 203, 1872, 11525, 41344, 68067, 31360, 985, 0, 1, 18, 264, 2940, 23364, 120750, 350272, 439992, 139536, 2378, 0

Offset: 0

Seiichi Manyama, Mar 01 2021

			Square array begins:
  1,  1,    1,     1,     1,      1, ...
  0,  2,    4,     6,     8,     10, ...
  0,  5,   18,    39,    68,    105, ...
  0, 12,   80,   252,   576,   1100, ...
  0, 29,  356,  1629,  4880,  11525, ...
  0, 70, 1584, 10530, 41344, 120750, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..5 give A000007, A000129(n+1), A090017(n+1), A090018, A190510(n+1), A190955(n+1).
Rows 0..2 give A000012, A005843, A007742.
Main diagonal gives A109517(n+1).
Cf. A342120, A342129, A342133.

T:= (n, k)-> (<<0|1>, >^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021

T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * 2^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)

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

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

T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I));

T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) + T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (1/2)^(n-j) * binomial(j,n-j).
T(n,k) = (-sqrt(k)*i)^n * U(n, sqrt(k)*i) where U(n, x) is a Chebyshev polynomial of the second kind.

cp's OEIS Frontend

A099089 Riordan array (1, 2+x).

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A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

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A342134 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx - k*x^2).

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cp's OEIS Frontend

A099089 Riordan array (1, 2+x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A342134 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x - k*x^2).

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Author

Keywords

Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A342134 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx - k*x^2).