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.

Previous Showing 11-16 of 16 results.

A168615 Inverse binomial transform of A169609, or of A144437 preceded by 1.

Original entry on oeis.org

1, 2, -2, 0, 6, -18, 36, -54, 54, 0, -162, 486, -972, 1458, -1458, 0, 4374, -13122, 26244, -39366, 39366, 0, -118098, 354294, -708588, 1062882, -1062882, 0, 3188646, -9565938, 19131876, -28697814, 28697814, 0, -86093442, 258280326, -516560652
Offset: 0

Views

Author

Paul Curtz, Dec 01 2009

Keywords

Links

Crossrefs

Cf. A169609, A144437, A000748, A057083, A057682.

Programs

  • Magma
    [ n le 2 select n else n eq 3 select -2 else -3*Self(n-1)-3*Self(n-2): n in [1..37] ]; // Klaus Brockhaus, Dec 03 2009
  • Mathematica
    Join[{1,2,-2}, LinearRecurrence[{-3, -3}, {0, 6}, 25]] (* G. C. Greubel, Jul 27 2016 *)
LinearRecurrence[{-3,-3},{1,2,-2},40] (* Harvey P. Dale, Jul 21 2024 *)

Formula

a(n) = -3*a(n-1) - 3*a(n-2) for n > 2; a(0) = 1, a(1) = 2, a(2) = -2.
a(n) = 2*A123877(n-1), n>0.
G.f.: 1+2*x*(1+2*x)/(1+3*x+3*x^2).
a(6*m + 3) = 0, m>=0. - G. C. Greubel, Jul 27 2016

Extensions

Edited and extended by Klaus Brockhaus, Dec 03 2009

A202209 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 13, 5, 0, 0, 34, 19, 1, 0, 0, 89, 65, 8, 0, 0, 0, 233, 210, 42, 1, 0, 0, 0, 610, 654, 183, 11, 0, 0, 0, 0, 1597, 1985, 717, 74, 1, 0, 0, 0, 0, 4181, 5911, 2622, 394, 14, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 14 2011

Keywords

Comments

Riordan array ((1-x)/(1-3x+x^2), x^2/(1-3x+x^2)) .

Examples

			Triangle begins :
1
2, 0
5, 1, 0
13, 5, 0, 0
34, 19, 1, 0, 0
89, 65, 8, 0, 0, 0
233, 210, 42, 1, 0, 0, 0

Crossrefs

Cf. A000045, A000079, A001519, A001870, A001906, A126124, A202207 (antidiagonal sums)

Formula

T(n,k) = 3*T(n-1,k) - T(n-2,k) + T(n-2,k-1).
G.f.: (1-x)/(1-3x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively.
T(n,0) = A122367(n) = A000045(2n+1).

A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980
Offset: 0

Views

Author

Paul Curtz, Jul 11 2013

Keywords

Comments

Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n): after 0, it is A192080.
0, 0, 0, 0, 1, 5, 15, 35, 70, 126,... e(n)
0, 0, 0, 1, 4, 10, 20, 35, 56, 85,... f(n)
0, 0, 1, 3, 6, 10, 15, 21, 29, 45,... a(n)
0, 1, 2, 3, 4, 5, 6, 8, 16, 45,... b(n)
1, 1, 1, 1, 1, 1, 2, 8, 29, 85,... c(n)
0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n).
a(n) + d(n) = A024495(n),
b(n) + e(n) = A131708(n),
c(n) + f(n) = A024493(n).
a(n) - d(n) = 0, 0, 1, 3, 6, 9, 9, 0,... A057083(n-2)
b(n) - e(n) = 0, 1, 2, 3, 3, 0, -9, -27,... A057682(n)
c(n) - f(n) = 1, 1, 1, 0, -3, -9, -18, -27,... A057681(n)
d(n) - a(n) = 0, 0, -1, -3, -6, -9, -9, 0,... -A057083(n-2)
e(n) - b(n) = 0, -1, -2, -3, -3, 0, 9, 27,... -A057682(n)
f(n) - c(n) = -1, -1, -1, 0, 3, 9, 18, 27,... -A057681(n).
The first column is A131531(n).
The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)?

Examples

			a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.

Links

Programs

  • Mathematica
    Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* Harvey P. Dale, Dec 17 2014 *)
  • PARI
    {a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019

Formula

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
a(n) = A024495(n) - A192080(n-5) for n>4.
G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013
a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019

Extensions

Definition uses the g.f. of Ralf Stephan.
More terms from Harvey P. Dale, Dec 17 2014

A183189 Triangle T(n,k), read by rows, given by (2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 6, 1, 0, 18, 5, 0, 0, 54, 21, 1, 0, 0, 162, 81, 8, 0, 0, 0, 486, 297, 45, 1, 0, 0, 0, 1458, 1053, 216, 11, 0, 0, 0, 0, 4374, 3645, 945, 78, 1, 0, 0, 0, 0, 13122, 12393, 3888, 450, 14, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 14 2011

Keywords

Comments

Riordan array ((1-x)/(1-3x), x^2/(1-3x)).
A skewed version of triangular array in A193723.
A202209*A007318 as infinite lower triangular matrices.

Examples

			Triangle begins:
  1
  2, 0
  6, 1, 0
  18, 5, 0, 0
  54, 21, 1, 0, 0
  162, 81, 8, 0, 0, 0
  486, 297, 45, 1, 0, 0, 0

Crossrefs

Cf. A000244, A025192, A081038, A183188 (antidiagonal sums).

Formula

G.f.: (1-x)/(1-3*x-y*x^2).
T(n,k) = Sum_{j, j>=0} T(n-2-j,k-1)*3^j.
T(n,k) = 3*T(n-1,k) + T(n-2,k-1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively.

A221179 A convolution triangle of numbers obtained from A146559.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, -2, 1, 3, 1, 0, -4, -4, 3, 4, 1, 0, -4, -12, -5, 6, 5, 1, 0, 0, -16, -24, -4, 10, 6, 1, 0, 8, -4, -42, -39, 0, 15, 7, 1, 0, 16, 32, -24, -88, -55, 8, 21, 8, 1, 0, 16, 80, 72, -80
Offset: 0

Views

Author

Philippe Deléham, Feb 20 2013

Keywords

Comments

Triangle T(n,k) given by (0, 1, -1, 2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Examples

			Triangle begins:
1
0, 1
0, 1, 1
0, 0, 2, 1
0, -2, 1, 3, 1
0, -4, -4, 3, 4, 1
0, -4, -12, -5, 6, 5, 1
0, 0, -16, -24, -4, 10, 6, 1

Crossrefs

Cf. A030523, A104597, A181472, A220399

Formula

G.f. for the k-th column: ((x-x^2)/(1-2*x+2*x^2))^k.
G.f.: (1-2*x+2*x^2)/(1-2*x+2*x^2-x*y+x^2*y).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = (-1)^(n-k)*A181472(n-1,k-1) for n>0 and k>0.
T(n,1) = A146559(n-1).
T(n+1,n) = n = A001477(n).
T(n+2,n) = (n^2-n)/2 = A161680(n).
Sum_{k, 0<=k<=n} T(n,k) = A057682(n) for n>0.

A236311 Riordan array ((1-x)/(1-3*x+3*x^2), x/(1-3*x+3*x^2)).

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 3, 15, 8, 1, 0, 33, 36, 11, 1, -9, 54, 117, 66, 14, 1, -27, 54, 297, 282, 105, 17, 1, -54, -27, 594, 945, 555, 153, 20, 1, -81, -297, 864, 2583, 2295, 963, 210, 23, 1, -81, -891, 513, 5778, 7803, 4725, 1533, 276, 26, 1, 0, -1863, -1944, 10098
Offset: 0

Views

Author

Philippe Deléham, Jan 21 2014

Keywords

Comments

Row sums are 3^n = A000244(n).
Diagonals sums are 2^n = A000079(n).
T(n,n) = A000012(n).
T(n+1,n) = A016789(n).
T(n+2,n) = A062741(n+1).
T(n+3,n) = 3*A004188(n+1).
T(n,0) = A057682(n+1).

Examples

			Triangle begins :
1;
2, 1;
3, 5, 1;
3, 15, 8, 1;
0, 33, 36, 11, 1;
-9, 54, 117, 66, 14, 1;
-27, 54, 297, 282, 105, 17, 1;

Crossrefs

Cf. A057083, A057682, A000012, A016789, A062741

Formula

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) -3*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k<0 or if k>n.
Previous Showing 11-16 of 16 results.

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