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.

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A028272 Elements to right of central elements in 3-Pascal triangle A028262 that are not 1.

Original entry on oeis.org

4, 5, 13, 6, 19, 7, 45, 26, 8, 71, 34, 9, 161, 105, 43, 10, 266, 148, 53, 11, 588, 414, 201, 64, 12, 1002, 615, 265, 76, 13, 2178, 1617, 880, 341, 89, 14, 3795, 2497, 1221, 430, 103, 15, 8151, 6292, 3718, 1651, 533, 118, 16, 14443, 10010, 5369, 2184, 651, 134
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028274 Odd elements (greater than 1) to right of central elements in 3-Pascal triangle A028262.

Original entry on oeis.org

5, 13, 19, 7, 45, 71, 9, 161, 105, 43, 53, 11, 201, 615, 265, 13, 1617, 341, 89, 3795, 2497, 1221, 103, 15, 8151, 1651, 533, 14443, 5369, 651, 17, 30745, 24453, 15379, 7553, 2835, 785, 151, 169, 19, 1105, 5661, 1293, 21, 24225, 1501, 229, 90117, 31179
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028275 Elements in 4-Pascal triangle (by row).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 10, 6, 1, 1, 7, 16, 16, 7, 1, 1, 8, 23, 32, 23, 8, 1, 1, 9, 31, 55, 55, 31, 9, 1, 1, 10, 40, 86, 110, 86, 40, 10, 1, 1, 11, 50, 126, 196, 196, 126, 50, 11, 1, 1, 12, 61, 176, 322, 392, 322, 176, 61, 12, 1, 1, 13, 73, 237, 498, 714, 714, 498, 237
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1; 1 1; 1 4 1; 1 5 5 1; 1 6 10 6 1; ...

Crossrefs

Cf. A028262, A028313.

Formula

Apart from first 3 rows, use the Pascal rule.
T(n, k) = C(n, k) + 2C(n-2, k-1). G.f.: (1+2x^2y) / [1-x(1+y)]. - Ralf Stephan, Jan 31 2005

Extensions

More terms from Ben Baugher (s1191623(AT)cedarville.edu)

A146986 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 12, 22, 12, 1, 1, 21, 58, 58, 21, 1, 1, 38, 143, 212, 143, 38, 1, 1, 71, 341, 675, 675, 341, 71, 1, 1, 136, 796, 1976, 2630, 1976, 796, 136, 1, 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1, 1, 522, 4141, 14456, 28882, 36092, 28882, 14456, 4141, 522, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 04 2008

Keywords

Comments

Row sums are: {1, 2, 6, 16, 48, 160, 576, 2176, 8448, 33280, 132096, ...} = {1, 2, 2*A242985(n)}. (modified by G. C. Greubel, Jan 09 2020)

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  4,   1;
  1,  7,   7,   1;
  1, 12,  22,  12,   1;
  1, 21,  58,  58,  21,  1;
  1, 38, 143, 212, 143, 38, 1;

Links

Crossrefs

Cf. A028262, A242985.

Programs

  • GAP
    T:= function(n,k,q)
    if n<2 then return Binomial(n,k);
    else return Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k,2) ))); # G. C. Greubel, Jan 09 2020
  • Magma
    T:= func< n,k,q | n lt 2 select Binomial(n,k) else Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1) >;
[T(n,k,2): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 09 2020
  • Maple
    q:=2; seq(seq( `if`(n<2, binomial(n,k), binomial(n,k) + q^(n-1)*binomial(n-2,k-1)), k=0..n), n=0..10); # G. C. Greubel, Jan 09 2020
  • Mathematica
    Table[If[n<2, Binomial[n, m], Binomial[n, m] + 2^(n-1)*Binomial[n-2, m-1]], {n, 0, 10}, {m, 0, n}]//Flatten
  • PARI
    T(n,k) = if(n<2, binomial(n,k), binomial(n,k) + 2^(n-1)*binomial(n-2,k-1) ); \\ G. C. Greubel, Jan 09 2020
  • Sage
    @CachedFunction
def T(n, k, q):
    if (n<2): return binomial(n,k)
    else: return binomial(n,k) + q^(n-1)*binomial(n-2,k-1)
[[T(n, k, 2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 09 2020

Formula

T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
Sum_{k=0..n} T(n,k) = n+1 for n < 2 and 16*binomial(2^(n-3) + 1, 2) otherwise. - G. C. Greubel, Jan 09 2020

Extensions

Edited by G. C. Greubel, Jan 09 2020

A146987 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 12, 12, 1, 1, 31, 60, 31, 1, 1, 86, 253, 253, 86, 1, 1, 249, 987, 1478, 987, 249, 1, 1, 736, 3666, 7325, 7325, 3666, 736, 1, 1, 2195, 13150, 32861, 43810, 32861, 13150, 2195, 1, 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 04 2008

Keywords

Comments

Row sums are: {1, 2, 7, 26, 124, 680, 3952, 23456, 140224, 840320, 5039872}.

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   5,   1;
  1,  12,  12,    1;
  1,  31,  60,   31,   1;
  1,  86, 253,  253,  86,   1;
  1, 249, 987, 1478, 987, 249, 1;

Links

Crossrefs

Cf. A028262.

Programs

  • GAP
    T:= function(n,k,q)
    if n<2 then return Binomial(n,k);
    else return Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1);
fi; end; Flat(List([0..10], n-> List([0..n], k-> T(n,k,3) ))); # G. C. Greubel, Jan 09 2020
  • Magma
    T:= func< n,k,q | n lt 2 select Binomial(n,k) else Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1) >;
[T(n,k,3): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 09 2020
  • Maple
    q:=3; seq(seq( `if`(n<2, binomial(n,k), binomial(n,k) + q^(n-1)*binomial(n-2,k-1)), k=0..n), n=0..10); # G. C. Greubel, Jan 09 2020
  • Mathematica
    Table[If[n<2, Binomial[n, m], Binomial[n, m] + 3^(n-1)*Binomial[n-2, m-1]], {n, 0, 10}, {m, 0, n}]//Flatten
  • PARI
    T(n,k) = if(n<2, binomial(n,k), binomial(n,k) + 3^(n-1)*binomial(n-2,k-1) ); \\ G. C. Greubel, Jan 09 2020
  • Sage
    @CachedFunction
def T(n, k, q):
    if (n<2): return binomial(n,k)
    else: return binomial(n,k) + q^(n-1)*binomial(n-2,k-1)
[[T(n, k, 3) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 09 2020

Formula

T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.

Extensions

Edited by G. C. Greubel, Jan 09 2020

A146988 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 4^(n-1) * binomial(n-2, k-1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 68, 134, 68, 1, 1, 261, 778, 778, 261, 1, 1, 1030, 4111, 6164, 4111, 1030, 1, 1, 4103, 20501, 40995, 40995, 20501, 4103, 1, 1, 16392, 98332, 245816, 327750, 245816, 98332, 16392, 1, 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 04 2008

Keywords

Comments

Row sums are {1, 2, 8, 40, 272, 2080, 16448, 131200, 1048832, 8389120, 67109888, ...} = {1, 2, 8*A081342(n)}. (modified by G. C. Greubel, Jan 09 2020)

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    6,    1;
  1,   19,   19,    1;
  1,   68,  134,   68,    1;
  1,  261,  778,  778,  261,    1;
  1, 1030, 4111, 6164, 4111, 1030, 1;

Links

Crossrefs

Cf. A028262, A081342.

Programs

  • GAP
    T:= function(n,k,q)
    if n<2 then return Binomial(n,k);
    else return Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k,4) ))); # G. C. Greubel, Jan 09 2020
  • Magma
    T:= func< n,k,q | n lt 2 select Binomial(n,k) else Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1) >;
[T(n,k,4): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 09 2020
  • Maple
    q:=4; seq(seq( `if`(n<2, binomial(n,k), binomial(n,k) + q^(n-1)*binomial(n-2,k-1)), k=0..n), n=0..10); # G. C. Greubel, Jan 09 2020
  • Mathematica
    Table[If[n<2, Binomial[n, m], Binomial[n, m] + 4^(n-1)*Binomial[n-2, m-1]], {n, 0, 10}, {m, 0, n}]//Flatten
  • PARI
    T(n,k) = if(n<2, binomial(n,k), binomial(n,k) + 4^(n-1)*binomial(n-2,k-1) ); \\ G. C. Greubel, Jan 09 2020
  • Sage
    @CachedFunction
def T(n, k, q):
    if (n<2): return binomial(n,k)
    else: return binomial(n,k) + q^(n-1)*binomial(n-2,k-1)
[[T(n, k, 4) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 09 2020

Formula

T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.
Sum_{k=0..n} T(n,k) = n+1 for n < 2 and 4*(2^n + 8^n) otherwise. - G. C. Greubel, Jan 09 2020

Extensions

Edited by G. C. Greubel, Jan 09 2020

A146990 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + n^(n-1) * binomial(n-2, k-1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 68, 134, 68, 1, 1, 630, 1885, 1885, 630, 1, 1, 7782, 31119, 46676, 31119, 7782, 1, 1, 117656, 588266, 1176525, 1176525, 588266, 117656, 1, 1, 2097160, 12582940, 31457336, 41943110, 31457336, 12582940, 2097160, 1
Offset: 0

Views

Author

Roger L. Bagula, Nov 04 2008

Keywords

Comments

Row sums are: {1, 2, 6, 26, 272, 5032, 124480, 3764896, 134217984, 5509980800, 256000001024, ...}.

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    4,     1;
  1,   12,    12,     1;
  1,   68,   134,    68,     1;
  1,  630,  1885,  1885,   630,    1;
  1, 7782, 31119, 46676, 31119, 7782, 1;

Links

Crossrefs

Cf. A028262.

Programs

  • GAP
    T:= function(n,k)
    if n<2 then return Binomial(n,k);
    else return Binomial(n,k) + n^(n-1)*Binomial(n-2,k-1);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jan 09 2020
  • Magma
    T:= func< n,k | n lt 2 select Binomial(n,k) else Binomial(n,k) + n^(n-1)*Binomial(n-2,k-1) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 09 2020
  • Maple
    seq(seq( `if`(n<2, binomial(n,k), binomial(n,k) + n^(n-1)*binomial(n-2,k-1)), k=0..n), n=0..10); # G. C. Greubel, Jan 09 2020
  • Mathematica
    Table[If[n <2, Binomial[n, m], Binomial[n, m] + n^(n - 1)*Binomial[n - 2, m - 1]], {n, 0, 10}, {m, 0, n}]; Flatten[%]
  • PARI
    T(n,k) = if(n<2, binomial(n,k), binomial(n,k) + n^(n-1)*binomial(n-2,k-1) ); \\ G. C. Greubel, Jan 09 2020
  • Sage
    @CachedFunction
def T(n, k):
    if (n<2): return binomial(n,k)
    else: return binomial(n,k) + n^(n-1)*binomial(n-2,k-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 09 2020

Formula

T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + n^(n-1) * binomial(n-2, k-1) otherwise.

Extensions

Edited by G. C. Greubel, Jan 09 2020

A051297 (Terms in A028266)/2.

Original entry on oeis.org

2, 2, 4, 3, 3, 13, 4, 13, 13, 4, 17, 45, 17, 5, 5, 74, 133, 161, 133, 74, 6, 32, 207, 294, 294, 207, 32, 6, 38, 501, 588, 501, 38, 7, 440, 1089, 1089, 440, 7, 215, 2178, 215, 8, 59, 1859, 3146, 3146, 1859, 59, 8, 67, 1092, 5005, 8151, 5005, 1092, 67, 9, 9, 468
Offset: 0

Views

Author

James Sellers

Keywords

Crossrefs

A028266, A028262.

A051298 (Terms in A028273)/2.

Original entry on oeis.org

2, 3, 13, 4, 17, 5, 133, 74, 294, 207, 32, 6, 501, 38, 1089, 440, 7, 215, 3146, 1859, 59, 8, 5005, 1092, 67, 9, 27599, 19916, 11466, 5194, 1810, 468, 58344, 47515, 31382, 16660, 7004, 2278, 94, 10, 105859, 78897, 48042, 23664, 9282, 104, 222547, 184756
Offset: 0

Views

Author

James Sellers

Keywords

Crossrefs

A028273, A028262.

A147644 Triangle read by rows: t(n,m)=Binomial[n, m] + If[n > 2, 2*Binomial[n - 2, m - 1], 0]; Mod[If[n > 2, 2*Binomial[n - 2, m - 1], 0],2]=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 6, 10, 6, 1, 1, 7, 16, 16, 7, 1, 1, 8, 23, 32, 23, 8, 1, 1, 9, 31, 55, 55, 31, 9, 1, 1, 10, 40, 86, 110, 86, 40, 10, 1, 1, 11, 50, 126, 196, 196, 126, 50, 11, 1, 1, 12, 61, 176, 322, 392, 322, 176, 61, 12
Offset: 0

Views

Author

Roger L. Bagula, Nov 09 2008

Keywords

Comments

Row sums are:{1, 2, 4, 12, 24, 48, 96, 192, 384, 768, 1536,...}

Examples

			{1}, {1, 1}, {1, 2, 1}, {1, 5, 5, 1}, {1, 6, 10, 6, 1}, {1, 7, 16, 16, 7, 1}, {1, 8, 23, 32, 23, 8, 1}, {1, 9, 31, 55, 55, 31, 9, 1}, {1, 10, 40, 86, 110, 86, 40, 10, 1}, {1, 11, 50, 126, 196, 196, 126, 50, 11, 1}, {1, 12, 61, 176, 322, 392, 322, 176, 61, 12, 1}

Crossrefs

A146986, A146987, A028262

Programs

  • Mathematica
    Table[Table[Binomial[n, m] + If[n > 2, 2*Binomial[n - 2, m - 1], 0], {m, 0, n}], {n, 0, 10}]; Flatten[%]
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