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.

Showing 1-4 of 4 results.

A147722 Row sums of Riordan array ((1-3x)/(1-4x+x^2), x(1-x)/(1-4x+x^2)).

Original entry on oeis.org

1, 2, 8, 36, 164, 748, 3412, 15564, 70996, 323852, 1477268, 6738636, 30738644, 140215948, 639602452, 2917580364, 13308696916, 60708323852, 276924225428, 1263204479436, 5762173946324, 26284460772748, 119897955971092, 546920858309964, 2494808379607636, 11380200181418252
Offset: 0

Views

Author

Paul Barry, Nov 11 2008

Keywords

Comments

Row sums of A147721.
Hankel transform of a(n) is [1,4,0,0,0,0,0,0,...]. [Philippe Deléham, Dec 03 2008]

Links

Programs

  • Mathematica
    CoefficientList[Series[(1 - 3x)/(1 - 5x + 2x^2) , {x, 0, 21}], x] (* Indranil Ghosh, Mar 10 2017 *)
LinearRecurrence[{5,-2},{1,2},30] (* Harvey P. Dale, Aug 26 2021 *)
  • PARI
    Vec((1 - 3*x)/(1 - 5*x + 2*x^2) + O(x^22)) \\ Indranil Ghosh, Mar 10 2017

Formula

G.f.: (1-3x)/(1-5x+2x^2).
a(n) = 5*a(n-1)-2*a(n-2), a(0)=1, a(1)=2. [Philippe Deléham, Nov 13 2008]

A147720 Riordan array (1, x(1-x)/(1-3x)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 18, 16, 6, 1, 0, 54, 60, 30, 8, 1, 0, 162, 216, 134, 48, 10, 1, 0, 486, 756, 558, 248, 70, 12, 1, 0, 1458, 2592, 2214, 1168, 410, 96, 14, 1, 0, 4374, 8748, 8478, 5160, 2150, 628, 126, 16
Offset: 0

Views

Author

Paul Barry, Nov 11 2008

Keywords

Comments

Array [0,2,1,0,0,0,....] DELTA [1,0,0,0,......] for Deléham DELTA as in A084938.
Row sums are A001835. Diagonal sums are related to A030186.
Row sums of inverse are essentially A091593. A147720*A007318 is A147721.

Examples

			Triangle begins
1;
0,   1;
0,   2,   1;
0,   6,   4,   1;
0,  18,  16,   6,   1;
0,  54,  60,  30,   8,   1;
0, 162, 216, 134,  48,  10,   1;

Links

Programs

  • Mathematica
    nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1-3*x)/(1-(3+y)*x+y*x^2), {x, 0, nmax}],x],{y,0,nmax}],y]] (* Indranil Ghosh, Mar 10 2017, after Philippe Deléham *)

Formula

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001835(n), A147722(n), A084120(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Nov 15 2008
G.f.: (1-3*x)/(1-(3+y)*x+y*x^2). - Philippe Deléham, Feb 15 2012

A147724 a(n) = C(3,n) DELTA C(0,n).

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 25, 33, 9, 1, 172, 238, 78, 13, 1, 1201, 1745, 667, 139, 17, 1, 8404, 12807, 5583, 1376, 216, 21, 1, 58825, 93841, 45822, 12950, 2429, 309, 25, 1, 411772, 686288, 370108, 117458, 25366, 3890, 418, 29, 1, 2882401, 5009889, 2951034, 1035834, 251583, 44607, 5823, 543, 33, 1
Offset: 0

Views

Author

Paul Barry, Nov 11 2008

Keywords

Comments

Triangle [1,3,3,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA as in A084938.
First column is A034494(n-1). Row sums are A147725. A147724 = A147723*A007318.

Examples

			Triangle begins
    1;
    1,   1;
    4,   5,   1;
   25,  33,   9,   1;
  172, 238,  78,  13,   1;

Links

Crossrefs

Cf. A147721.

Programs

  • Mathematica
    nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - 7*x + 3*x^2)/(1 - 8*x + 7*x^2 - x*y + 4*x^2*y) , {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017 *)

Formula

Riordan array ((1-7*x+3*x^2)/(1-8*x+7*x^2), x*(1-4*x)/(1-8*x+7*x^2)).
G.f.: (1 - 7*x + 3*x^2)/(1 - 8*x + 7*x^2 - x*y + 4*x^2*y). - Philippe Deléham, Oct 29 2013
T(n,k) = 8*T(n-1,k) + T(n-1,k-1) - 7*T(n-2,k) - 4*T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 4, T(2,1) = 5, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 29 2013

A206831 Triangle T(n,k), read by rows, given by (1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, -1, -3, -1, 1, 1, 0, -4, -2, 1, 1, 5, 4, -4, -3, 1, -1, 0, 9, 10, -3, -4, 1, -1, -7, -9, 9, 17, -1, -5, 1, 1, 0, -16, -28, 2, 24, 2, -6, 1, 1, 9, 16, -16, -54, -14, 30, 6, -7, 1, -1, 0, 25, 60, 10
Offset: 0

Views

Author

Philippe Deléham, Feb 13 2012

Keywords

Comments

Riordan array ((1+x)/(1+x^2), x*(1-x)/(1+x^2)).
Antidiagonal sums are A010892(n).

Examples

			Triangle begins :
1
1, 1
-1, 0, 1
-1, -3, -1, 1
1, 0, -4, -2, 1
1, 5, 4, -4, -3, 1
-1, 0, 9, 10, -3, -4, 1
-1, -7, -9, 9, 17, -1, -5, 1
1, 0, -16, -28, 2, 24, 2, -6, 1
1, 9, 16, -16, -54, -14, 30, 6, -7, 1
-1, 0, 25, 60, 10, -80, -40, 34, 11, -8, 1

Links

Crossrefs

Cf. A129267, A007318, A147703, A147721
Cf. A057077, A087960

Programs

  • Mathematica
    nmax=10; Flatten[CoefficientList[Series[CoefficientList[Series[(1 + x)/(1 - y*x + (1 + y)*x^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017 *)

Formula

T(n,k) = T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), n>1.
G.f.: (1+x)/(1-y*x+(1+y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A057077(n), (-1)^n*A078050(n) for x = -1, 0, 1 respectively.
Showing 1-4 of 4 results.

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