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 1-10 of 10 results.

A054091 Row sums of A054090.

Original entry on oeis.org

1, 2, 4, 10, 32, 130, 652, 3914, 27400, 219202, 1972820, 19728202, 217010224, 2604122690, 33853594972, 473950329610, 7109254944152, 113748079106434, 1933717344809380, 34806912206568842, 661331331924808000, 13226626638496160002, 277759159408419360044
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Row sums of A094816 as a triangular array as follows: {1}, {1,1}, {1,3}, {1,1,8}, {6,1,1,24}, {29,10,1,1,89}, ... - Michael Somos, Nov 19 2006
a(n) = (n-1)a(n-1)+2, n>0; 2=0*1+2, 4=1*2+2, 10=2*4+2, ... - Gary Detlefs, May 20 2010
Row sums of triangle A208058. - Gary W. Adamson, Feb 22 2012

Links

Crossrefs

Cf. A000522, A054090, A094816, A208058.

Programs

  • Magma
    [n le 2 select n else (n-1)*Self(n-1) -(n-3)*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 23 2022
  • Maple
    a:= n-> `if`( n=0, 1, add(2*(n-1)!/j!, j=0..n-1)): seq(a(n), n=0..18); # Zerinvary Lajos, Oct 20 2006
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, 2+(n-1)*a(n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 23 2022
  • Mathematica
    Table[If[n==0, 1, 2*(n-1)!*Sum[1/j!, {j,0,n-1}]], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)
  • PARI
    {a(n)= local(A); if(n<1, n==0, A=vector(n); A[1]=2; for(k=1, n-1, A[k+1]=k*A[k]+2); A[n])} /* Michael Somos, Nov 19 2006 */
  • PARI
    {a(n)= if(n<1, n==0, n--; n!*polcoeff( 2*exp(x+x*O(x^n))/(1-x), n))} /* Michael Somos, Nov 19 2006 */
  • SageMath
    [1]+[2*factorial(n-1)*sum(1/factorial(j) for j in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jun 23 2022

Formula

a(n+1) = 2*A000522(n).
a(n+1) = Sum(2*n!/j!, j=0..n). - Zerinvary Lajos, Oct 20 2006
a(n) = 2*floor(e*(n-1)!), n>1. - Gary Detlefs, May 20 2010
a(n) = n*a(n-1) - (n-2)*a(n-2), a(0)=1, a(1)=2. - Vincenzo Librandi, Feb 23 2012

A054096 T(n,2), array T as in A054090.

Original entry on oeis.org

1, 2, 6, 22, 98, 522, 3262, 23486, 191802, 1753618, 17755382, 197282022, 2387112466, 31249472282, 440096734638, 6635304614542, 106638824162282, 1819969265702946
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

Conjecture: sequence is A006183 shifted right. - Ralf Stephan, Jan 15 2004

A054092 T(n,n), array T as in A054090.

Original entry on oeis.org

1, 1, 1, 3, 7, 25, 105, 547, 3367, 24033, 195169, 1777651, 17950551, 199059673, 2405063017, 31448531955, 442501797655, 6666753146497, 107081325959937, 1826636018849443, 32980276187719399, 628351055737088601
Offset: 0

Views

Author

Clark Kimberling

Keywords

Programs

  • PARI
    {a(n)= local(A); if(n<3, n>=0, A=vector(n, i, 1); for(k=1, n-2, A[k+2]=(k-1)*A[k+1]+ k*A[k]+ 2); A[n])} /* Michael Somos, Nov 19 2006 */

A054093 T(n,n-1), array T as in A054090.

Original entry on oeis.org

1, 2, 2, 8, 24, 106, 546, 3368, 24032, 195170, 1777650, 17950552, 199059672, 2405063018, 31448531954, 442501797656, 6666753146496, 107081325959938, 1826636018849442
Offset: 1

Views

Author

Clark Kimberling

Keywords

A054094 T(n,n-2), array T as in A054090.

Original entry on oeis.org

1, 4, 6, 26, 104, 548, 3366, 24034, 195168, 1777652, 17950550, 199059674, 2405063016, 31448531956, 442501797654, 6666753146498, 107081325959936, 1826636018849444
Offset: 2

Views

Author

Clark Kimberling

Keywords

A054095 T(n,n-3), array T as in A054090.

Original entry on oeis.org

1, 10, 22, 108, 544, 3370, 24030, 195172, 1777648, 17950554, 199059670, 2405063020, 31448531952, 442501797658, 6666753146494, 107081325959940, 1826636018849440
Offset: 3

Views

Author

Clark Kimberling

Keywords

A054097 T(n,3), array T as in A054090.

Original entry on oeis.org

3, 8, 26, 108, 554, 3392, 24138, 195716, 1781018, 17974584, 199254842, 2406840668, 31466482506, 442700857328, 6669158209514, 107112774491892, 1827078520647098
Offset: 3

Views

Author

Clark Kimberling

Keywords

A156184 A generalized recursion triangle sequence : m=1; t(n,k)=(k + m - 1)*t(n - 1, k, m) + (m*n - k + 1 - m)*t(n - 1, k - 1, m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 16, 7, 1, 1, 11, 53, 53, 11, 1, 1, 16, 150, 318, 150, 16, 1, 1, 22, 380, 1554, 1554, 380, 22, 1, 1, 29, 892, 6562, 12432, 6562, 892, 29, 1, 1, 37, 1987, 25038, 82538, 82538, 25038, 1987, 37, 1, 1, 46, 4270, 89023, 480380, 825380, 480380
Offset: 0

Views

Author

Roger L. Bagula, Feb 05 2009

Keywords

Comments

Row sums are: A054091;
{1, 2, 4, 10, 32, 130, 652, 3914, 27400, 219202, 1972820, ...}.
The sequence comes from a generalization of the recurrence for A008517.

Examples

			{1},
{1, 1},
{1, 2, 1},
{1, 4, 4, 1},
{1, 7, 16, 7, 1},
{1, 11, 53, 53, 11, 1},
{1, 16, 150, 318, 150, 16, 1},
{1, 22, 380, 1554, 1554, 380, 22, 1},
{1, 29, 892, 6562, 12432, 6562, 892, 29, 1},
{1, 37, 1987, 25038, 82538, 82538, 25038, 1987, 37, 1},
{1, 46, 4270, 89023, 480380, 825380, 480380, 89023, 4270, 46, 1}

Links

Crossrefs

Cf. A054091, A054090, A008517, A156141.

Programs

  • Mathematica
    m = 1; e[n_, 0, m_] := 1;
e[n_, k_, m_] := 0 /; k >= n;
e[n_, k_, 1] := 1 /; k >= n;
e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];
Table[Table[e[n, k, m], {k, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,k) = (k + m - 1)*t(n - 1, k, m) + (m*n - k + 1 - m)*t(n - 1, k - 1, m).

A156186 Triangle: m=3; e(n,k,n) = (k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m); t(n,k) = e(n,k,m) + e(n,n-k,m).

Original entry on oeis.org

2, 1, 1, 1, 6, 1, 1, 30, 30, 1, 1, 159, 360, 159, 1, 1, 1119, 3639, 3639, 1119, 1, 1, 10932, 41262, 57414, 41262, 10932, 1, 1, 136764, 582642, 898632, 898632, 582642, 136764, 1, 1, 2031933, 9957168, 16634718, 17182152, 16634718, 9957168, 2031933, 1, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 05 2009

Keywords

Examples

			{2},
{1, 1},
{1, 6, 1},
{1, 30, 30, 1},
{1, 159, 360, 159, 1},
{1, 1119, 3639, 3639, 1119, 1},
{1, 10932, 41262, 57414, 41262, 10932, 1},
{1, 136764, 582642, 898632, 898632, 582642, 136764, 1},
{1, 2031933, 9957168, 16634718, 17182152, 16634718, 9957168, 2031933, 1},...

Crossrefs

Cf. A054091, A054090, A008517, A156141.

Programs

  • Mathematica
    m = 3; e[n_, 0, m_] := 1;
e[n_, k_, m_] := 0 /; k >= n;
e[n_, k_, 1] := 1 /; k >= n;
e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];
Table[Table[e[n, k, m], {k, 0, n - 1}], {n, 1, 10}];
Table[Table[e[n, k, m] + e[n, n - k, m], {k, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

m=3; e(n,k,n) = (k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m);
t(n,k) = e(n,k,m) + e(n,n-k,m).

A156188 Triangle: m=5; e(n,k,n)=(k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m); t(n,k)=e(n,k,m)+e(n,n-k,m).

Original entry on oeis.org

2, 1, 1, 1, 10, 1, 1, 80, 80, 1, 1, 775, 1520, 775, 1, 1, 10915, 25945, 25945, 10915, 1, 1, 213720, 542910, 624670, 542910, 213720, 1, 1, 5245530, 14690640, 16408670, 16408670, 14690640, 5245530, 1, 1, 151534685, 479956020, 553630850, 464654480
Offset: 0

Views

Author

Roger L. Bagula, Feb 05 2009

Keywords

Examples

			{2},
{1, 1},
{1, 10, 1},
{1, 80, 80, 1},
{1, 775, 1520, 775, 1},
{1, 10915, 25945, 25945, 10915, 1},
{1, 213720, 542910, 624670, 542910, 213720, 1},
{1, 5245530, 14690640, 16408670, 16408670, 14690640, 5245530, 1},...

Crossrefs

Cf. A054091, A054090, A008517, A156141.

Programs

  • Mathematica
    m = 5; e[n_, 0, m_] := 1;
e[n_, k_, m_] := 0 /; k >= n;
e[n_, k_, 1] := 1 /; k >= n;
e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];
Table[Table[e[n, k, m], {k, 0, n - 1}], {n, 1, 10}];
Table[Table[e[n, k, m] + e[n, n - k, m], {k, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

m=5; e(n,k,n) = (k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m);
t(n,k) = e(n,k,m) + e(n,n-k,m).
Showing 1-10 of 10 results.

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