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 1-5 of 5 results.

A048487 a(n) = T(4,n), array T given by A048483.

Original entry on oeis.org

1, 6, 16, 36, 76, 156, 316, 636, 1276, 2556, 5116, 10236, 20476, 40956, 81916, 163836, 327676, 655356, 1310716, 2621436, 5242876, 10485756, 20971516, 41943036, 83886076, 167772156, 335544316, 671088636, 1342177276, 2684354556, 5368709116, 10737418236, 21474836476
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Row sums of triangle A131113. - Gary W. Adamson, Jun 15 2007
a(n) = sum of (n+1)-th row terms of triangle A134636. This sequence is the binomial transform of 1, 5, 5, (5 continued). - Gary W. Adamson, Nov 04 2007
Row sums of triangle A135856. - Gary W. Adamson, Dec 01 2007

Links

Crossrefs

Cf. A010716 (n-th difference of a(n), a(n-1), ..., a(0)).
Diagonal of A062001.
A column of A119726.
Cf. A048483, A123208, A131113, A134636, A135856.

Programs

Formula

a(n) = 5*2^n - 4. - Henry Bottomley, May 29 2001
a(n) = 2*a(n-1) + 4 for n > 0 with a(0) = 1. - Paul Barry, Aug 25 2004
From Colin Barker, Sep 13 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n >= 2.
G.f.: (1 + 3*x)/((1 - x)*(1 - 2*x)). (End)
a(n) = A123208(2*n). - Philippe Deléham, Apr 15 2013
E.g.f.: exp(x)*(5*exp(x) - 4). - Stefano Spezia, Oct 03 2023

A131115 Triangle read by rows: T(n,k) = 7*binomial(n,k) for 1 <= k <= n with T(n,n) = 1 for n >= 0.

Original entry on oeis.org

1, 7, 1, 7, 14, 1, 7, 21, 21, 1, 7, 28, 42, 28, 1, 7, 35, 70, 70, 35, 1, 7, 42, 105, 140, 105, 42, 1, 7, 49, 147, 245, 245, 147, 49, 1, 7, 56, 196, 392, 490, 392, 196, 56, 1, 7, 63, 252, 588, 882, 882, 588, 252, 63, 1, 7, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 15 2007

Keywords

Comments

Row sums give A048489.
Non-diagonal entries of Pascal's triangle are multiplied by 7. - Emeric Deutsch, Jun 20 2007
The matrix inverse starts
1;
-7, 1;
91, -14, 1;
-1771, 273, -21, 1;
45955, -7084, 546, -28, 1;
-1490587, 229775, -17710, 910, -35, 1;
58018051, -8943522, 689325, -35420, 1365, -42, 1;
-2634606331, 406126357, -31302327, 1608425, -61985, 1911, -49, 1;
... - R. J. Mathar, Mar 15 2013

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  7,  1;
  7, 14,  1;
  7, 21, 21,  1;
  7, 28, 42, 28,  1;
  7, 35, 70, 70, 35, 1;
  ...

Links

Crossrefs

Cf. A007318, A048489, A131110, A131111, A131112, A131113, A131114.

Programs

  • GAP
    T:= function(n,k)
    if k=n then return 1;
    else return 7*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019
  • Magma
    [k eq n select 1 else 7*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019
  • Maple
    T := proc (n, k) if k < n then 7*binomial(n, k) elif k = n then 1 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 20 2007
  • Mathematica
    Table[If[k==n, 1, 7*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)
  • PARI
    T(n,k)=if(k==n,1,7*binomial(n,k)) \\ Charles R Greathouse IV, Jan 16 2012
  • Sage
    @CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 7*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

Formula

G.f.: (1 + 6*x - t*x)/((1-t*x)*(1-x-t*x)). - Emeric Deutsch, Jun 20 2007

Extensions

Corrected and extended by Emeric Deutsch, Jun 20 2007

A131112 T(n,k) = 4*binomial(n,k) - 3*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 4, 1, 4, 8, 1, 4, 12, 12, 1, 4, 16, 24, 16, 1, 4, 20, 40, 40, 20, 1, 4, 24, 60, 80, 60, 24, 1, 4, 28, 84, 140, 140, 84, 28, 1, 4, 32, 112, 224, 280, 224, 112, 32, 1, 4, 36, 144, 336, 504, 504, 336, 144, 36, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 15 2007

Keywords

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  4,  1;
  4,  8,  1;
  4, 12, 12,  1;
  4, 16, 24, 16,  1;
  4, 20, 40, 40, 20, 1;
  ...

Links

Crossrefs

Cf. A007318, A036563, A131110, A131111, A131113, A131114, A131115.

Programs

  • GAP
    T:= function(n,k)
    if k=n then return 1;
    else return 4*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019
  • Magma
    [k eq n select 1 else 4*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019
  • Maple
    seq(seq(`if`(k=n, 1, 4*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019
  • Mathematica
    Table[If[k==n, 1, 4*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)
  • PARI
    T(n,k) = if(k==n, 1, 4*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019
  • Sage
    def T(n, k):
    if (k==n): return 1
    else: return 4*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)]
# G. C. Greubel, Nov 18 2019

Formula

T(n,k) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
n-th row sum = A036563(n+2) = 2^(n+2) - 3.
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 3*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021

A131114 T(n,k) = 6*binomial(n,k) - 5*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 6, 1, 6, 12, 1, 6, 18, 18, 1, 6, 24, 36, 24, 1, 6, 30, 60, 60, 30, 1, 6, 36, 90, 120, 90, 36, 1, 6, 42, 126, 210, 210, 126, 42, 1, 6, 48, 168, 336, 420, 336, 168, 48, 1, 6, 54, 216, 504, 756, 756, 504, 216, 54, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 15 2007

Keywords

Comments

Row sums give A048488.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  6,  1;
  6, 12,  1;
  6, 18, 18,   1;
  6, 24, 36,  24,  1;
  6, 30, 60,  60, 30,  1;
  6, 36, 90, 120, 90, 36, 1;
  ...

Links

Crossrefs

Cf. A007318, A048488, A131110, A131111, A131112, A131113, A131115.

Programs

  • GAP
    T:= function(n,k)
    if k=n then return 1;
    else return 6*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019
  • Magma
    [k eq n select 1 else 6*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019
  • Maple
    seq(seq(`if`(k=n, 1, 6*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019
  • Mathematica
    Table[If[k==n, 1, 6*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)
  • PARI
    T(n,k) = if(k==n, 1, 6*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019
  • Sage
    def T(n, k):
    if (k==n): return 1
    else: return 6*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

Formula

T(n,k) = 6*A007318(n,k) - 5*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 5*x - x*y)/((1 - x*y)*(1 - x - x*y)).

A131111 T(n, k) = 3*binomial(n,k) - 2*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

Original entry on oeis.org

1, 3, 1, 3, 6, 1, 3, 9, 9, 1, 3, 12, 18, 12, 1, 3, 15, 30, 30, 15, 1, 3, 18, 45, 60, 45, 18, 1, 3, 21, 63, 105, 105, 63, 21, 1, 3, 24, 84, 168, 210, 168, 84, 24, 1, 3, 27, 108, 252, 378, 378, 252, 108, 27, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 15 2007

Keywords

Comments

Row sums = A033484: (1, 4, 10, 22, 46, ...) = 3*2^n - 2.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  3,  1;
  3,  6,  1;
  3,  9,  9,  1;
  3, 12, 18, 12,  1;
  3, 15, 30, 30, 15,  1;
  3, 18, 45, 60, 45, 18, 1;
  ...

Links

Crossrefs

Cf. A131110, A131112, A131113, A131114, A131115.

Programs

  • GAP
    T:= function(n,k)
    if k=n then return 1;
    else return 3*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019
  • Magma
    [k eq n select 1 else 3*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019
  • Maple
    seq(seq(`if`(k=n, 1, 3*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019
  • Mathematica
    Table[If[k==n, 1, 3*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)
  • PARI
    T(n,k) = if(k==n, 1, 3*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019
  • Sage
    @CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 3*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

Formula

T(n,k) = 3*A007318(n,k) - 2*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 2*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.