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.

A174673 Triangle read by rows: T(n,m)=A154694(n,m)-A154694(n,0)+1.

Original entry on oeis.org

1, 1, 1, 1, 36, 1, 1, 296, 296, 1, 1, 1932, 4656, 1932, 1, 1, 11696, 54086, 54086, 11696, 1, 1, 69048, 556596, 1042920, 556596, 69048, 1, 1, 405236, 5406866, 16866206, 16866206, 5406866, 405236, 1, 1, 2381700, 51004320, 247754256, 404837664
Offset: 0

Views

Author

Roger L. Bagula, Mar 26 2010

Keywords

Comments

Reduces the values in the triangle A154694 such that each row starts with 1.
Row sums are:
1, 2, 38, 594, 8522, 131566, 2294210, 45356618, 1007118218, 24839902470,
673894929842,...

Examples

			{1},
{1, 1},
{1, 36, 1},
{1, 296, 296, 1},
{1, 1932, 4656, 1932, 1},
{1, 11696, 54086, 54086, 11696, 1},
{1, 69048, 556596, 1042920, 556596, 69048, 1},
{1, 405236, 5406866, 16866206, 16866206, 5406866, 405236, 1},
{1, 2381700, 51004320, 247754256, 404837664, 247754256, 51004320, 2381700, 1},
{1, 14050376, 473595806, 3441231326, 8491073726, 8491073726, 3441231326, 473595806, 14050376, 1},
{1, 83216400, 4357421004, 46167420504, 164067684600, 244543444824, 164067684600, 46167420504, 4357421004, 83216400, 1}

Crossrefs

A154690, A154692, A154693, A154964

Programs

  • Maple
    A174673 := proc(n,m)
    A154694(n,m)-A154694(n,0)+1 ;
end proc:
seq(seq( A174673(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Mar 11 2024
  • Mathematica
    Clear[t, p, q, n, m];
p = 2; q = 3;
t[n_, m_] = (p^(n - m)*q^m + p^m*q^( n - m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}];
Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,m)=A154694(n,m)-A154694(n,0)+1

A154693 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).

Original entry on oeis.org

2, 3, 3, 5, 16, 5, 9, 66, 66, 9, 17, 260, 528, 260, 17, 33, 1026, 3624, 3624, 1026, 33, 65, 4080, 23820, 38656, 23820, 4080, 65, 129, 16302, 154548, 374856, 374856, 154548, 16302, 129, 257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260, 257
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

Keywords

Comments

From G. C. Greubel, Jan 17 2025: (Start)
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A008292(n+1, k+1). When (p, q) = (2, 1) this sequence is obtained.
Some related triangles are:
(p, q) = (1, 1) : 2*A008292(n,k).
(p, q) = (2, 2) : 2*A257609(n,k).
(p, q) = (3, 2) : A154694(n,k).
(p, q) = (3, 3) : 2*A257620(n,k). (End)

Examples

			The triangle begins as:
    2;
    3,     3;
    5,    16,      5;
    9,    66,     66,       9;
   17,   260,    528,     260,      17;
   33,  1026,   3624,    3624,    1026,      33;
   65,  4080,  23820,   38656,   23820,    4080,     65;
  129, 16302, 154548,  374856,  374856,  154548,  16302,   129;
  257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260,  257;

Links

Crossrefs

Cf. A000629 (row sums), A008292, A154694, A257609, A257620.

Programs

  • Magma
    A154693:= func< n,k | (2^(n-k) + 2^k)*EulerianNumber(n+1, k) >;
[A154693(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025
  • Mathematica
    p=2; q=1;
A008292[n_,k_]:= A008292[n,k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
T[n_, m_]:= (p^(n-m)*q^m + p^m*q^(n-m))*A008292[n+1,m+1];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten
  • SageMath
    from sage.combinat.combinat import eulerian_number
def A154693(n,k): return (2^(n-k) +2^k)*eulerian_number(n+1,k)
print(flatten([[A154693(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 17 2025

Formula

T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1)
Sum_{k=0..n} T(n, k) = A000629(n+1).

Extensions

Definition simplified by the Assoc. Eds. of the OEIS - Aug 08 2010.

A174674 Sequence A154695 adjusted to leading one:t(n,m)=A154695(n,m)-A154695(n,0)+1.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 130, 130, 1, 1, 744, 1824, 744, 1, 1, 4234, 20152, 20152, 4234, 1, 1, 24484, 210796, 376704, 210796, 24484, 1, 1, 143686, 2183524, 6233224, 6233224, 2183524, 143686, 1, 1, 851504, 22549360, 99411264, 149600192, 99411264
Offset: 0

Views

Author

Roger L. Bagula, Mar 26 2010

Keywords

Comments

Row sums are:
1, 2, 22, 262, 3314, 48774, 847266, 17120870, 395224450, 10263445126,
296140564130,...

Examples

			{1},
{1, 1},
{1, 20, 1},
{1, 130, 130, 1},
{1, 744, 1824, 744, 1},
{1, 4234, 20152, 20152, 4234, 1},
{1, 24484, 210796, 376704, 210796, 24484, 1},
{1, 143686, 2183524, 6233224, 6233224, 2183524, 143686, 1},
{1, 851504, 22549360, 99411264, 149600192, 99411264, 22549360, 851504, 1},
{1, 5075122, 231836368, 1562973472, 3331837600, 3331837600, 1562973472, 231836368, 5075122, 1},
{1, 30344508, 2370195636, 24248921920, 72553861536, 97733916928, 72553861536, 24248921920, 2370195636, 30344508, 1}

Crossrefs

A154690, A154692, A154693, A154694, A154695

Programs

  • Mathematica
    Clear[t, p, q, n, m, a];
p[x_, n_] = 2^n*(1 - x)^(n + 1)*LerchPhi[x, -n, 1/2];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
p = 2; q = 1;
t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*a[[n + 1]][[m + 1]];
Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,m)=A154695(n,m)-A154695(n,0)+1

A174675 Sequence A154696 adjusted to leading one:t(n,m)=A154696(n,m)-A154696(n,0)+1.

Original entry on oeis.org

1, 1, 1, 1, 60, 1, 1, 656, 656, 1, 1, 5832, 16464, 5832, 1, 1, 49496, 302486, 302486, 49496, 1, 1, 419412, 4933332, 10171944, 4933332, 419412, 1, 1, 3593036, 76425506, 280498526, 280498526, 76425506, 3593036, 1, 1, 31167600, 1157982288
Offset: 0

Views

Author

Roger L. Bagula, Mar 26 2010

Keywords

Comments

Row sums are:
1, 2, 62, 1314, 28130, 703966, 20877434, 721034138, 28453293026,
1263142713270, 62305874244266,...

Examples

			{1},
{1, 1},
{1, 60, 1},
{1, 656, 656, 1},
{1, 5832, 16464, 5832, 1},
{1, 49496, 302486, 302486, 49496, 1},
{1, 419412, 4933332, 10171944, 4933332, 419412, 1},
{1, 3593036, 76425506, 280498526, 280498526, 76425506, 3593036, 1},
{1, 31167600, 1157982288, 6978681888, 12117629472, 6978681888, 1157982288, 31167600, 1},
{1, 273237776, 17387745806, 164112248126, 449798124926, 449798124926, 164112248126, 17387745806, 273237776, 1},
{1, 2414712204, 260247533196, 3735760480536, 15279843395064, 23749342002264, 15279843395064, 3735760480536, 260247533196, 2414712204, 1}

Crossrefs

A154690, A154692, A154693, A154694, A154695, A154696

Programs

  • Mathematica
    Clear[t, p, q, n, m, a];
p[x_, n_] = 2^n*(1 - x)^(n + 1)*LerchPhi[x, -n, 1/2];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
p = 2; q = 3;
t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*a[[n + 1]][[m + 1]];
Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,m)=A154696(n,m)-A154696(n,0)+1
Showing 1-4 of 4 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.