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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A225415 Triangle read by rows: absolute values of odd-numbered rows of A225434.

Original entry on oeis.org

1, 1, 58, 1, 1, 1556, 12006, 1556, 1, 1, 39054, 1461615, 5647300, 1461615, 39054, 1, 1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1, 1, 24414050, 11462721645, 414730580760, 3221733789330, 6783391017228, 3221733789330, 414730580760, 11462721645, 24414050, 1
Offset: 1

Views

Author

Roger L. Bagula, May 07 2013

Keywords

Examples

			Triangle begins:
  1;
  1,     58,         1;
  1,   1556,     12006,       1556,          1;
  1,  39054,   1461615,    5647300,    1461615,      39054,         1;
  1, 976552, 135028828, 1838120344, 4873361350, 1838120344, 135028828, 976552, 1;

Links

Crossrefs

The m=4 triangle in the sequence A034870 (m=0), A171692 (m=1), A225076 (m=2), A225398 (m=3).

Programs

  • Mathematica
    (* First program *)
t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1,(m*n-m*k+1)*t[n-1, k-1, m] + (m*k-(m-1))*t[n-1, k, m]];
T[n_, k_]:= T[n, k] = t[n+1, k+1,4]; (* t(n,k,4) = A142459 *)
Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(1+x), x], {n,1,14,2}]]
(* Second program *)
t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k-m+1)*t[n-1,k,m]]; (* t(n,k,4) = A142459 *)
T[n_, k_]:= T[n, k]= Sum[ (-1)^(k-j-1)*t[2*n,j+1,4], {j,0,k-1}];
Table[T[n, k], {n,12}, {k,2*n-1}]//Flatten (* G. C. Greubel, Mar 19 2022 *)
  • Sage
    @CachedFunction
def T(n, k, m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A142459(n, k): return T(n, k, 4)
def A225415(n,k): return sum( (-1)^(k-j-1)*A142459(2*n, j+1) for j in (0..k-1) )
flatten([[A225415(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022

Formula

T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142459(2*n, j+1). - G. C. Greubel, Mar 19 2022

Extensions

Edited by N. J. A. Sloane, May 11 2013

A152971 A vector sequence with set row sum function: row(n)=(2*n)!/n! and linear build up and decline function: f(n,m)=Floor[(m/n)*row(n)].

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 59, 59, 1, 1, 210, 1258, 210, 1, 1, 3024, 12095, 12095, 3024, 1, 1, 55440, 110880, 332638, 110880, 55440, 1, 1, 1235520, 2471040, 4942079, 4942079, 2471040, 1235520, 1, 1, 32432400, 64864800, 97297200, 129729598, 97297200
Offset: 0

Views

Author

Roger L. Bagula, Dec 16 2008

Keywords

Comments

row sums (2*n)!/n!:
{1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 67044257280,...}

Examples

			{1},
{1, 1},
{1, 10, 1},
{1, 59, 59, 1},
{1, 210, 1258, 210, 1},
{1, 3024, 12095, 12095, 3024, 1},
{1, 55440, 110880, 332638, 110880, 55440, 1},
{1, 1235520, 2471040, 4942079, 4942079, 2471040, 1235520, 1},
{1, 32432400, 64864800, 97297200, 129729598, 97297200, 64864800, 32432400, 1},
{1, 980179200, 1960358400, 2940537600, 2940537599, 2940537599, 2940537600, 1960358400, 980179200, 1},
{1, 33522128640, 67044257280, 100566385920, 134088514560, -2, 134088514560, 100566385920, 67044257280, 33522128640, 1}

Crossrefs

A142459

Programs

  • Mathematica
    Clear[v, n, row, f]; row[n_] = (2*n)!/n!;
f[n_, m_] = Floor[(m/n)*row[n]/2]; v[0] = {1}; v[1] = {1, 1};
v[n_] := v[n] = If[Mod[n, 2] == 0, Join[{1}, Table[ f[n, m], {m, 1, Floor[ n/2] - 1}], {row[n] - 2*Sum[ f[n, m], {m, 1, Floor[n/2] - 1}] - 2}, Table[ f[n, m], {m, Floor[n/ 2] - 1, 1, -1}], { 1}],
Join[{1}, Table[ f[n, m], {m, 1, Floor[n/2] - 1}], {row[n]/2 - Sum[ f[n, m], { m, 1, Floor[n/2] - 1}] - 1, row[n]/ 2 - Sum[ f[n, m], {m, 1, Floor[ n/2] - 1}] - 1}, Table[ f[n, m], {m, Floor[n/ 2] - 1, 1, -1}], {1}]];
Table[v[n], {n, 0, 10}]; Flatten[%]

Formula

row(n)=(2*n)!/n!: f(n,m)=Floor[(m/n)*row(n)].

A152972 A vector sequence with set row sum function: row(n)=-Product[3*k - 1, {k, 0, n}] and linear build up and decline function: f(n,m)=Floor[(m/n)*row(n)].

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 110, 658, 110, 1, 1, 1232, 4927, 4927, 1232, 1, 1, 17453, 34906, 104720, 34906, 17453, 1, 1, 299200, 598400, 1196799, 1196799, 598400, 299200, 1, 1, 6021400, 12042800, 18064200, 24085598, 18064200, 12042800
Offset: 0

Views

Author

Roger L. Bagula, Dec 16 2008

Keywords

Comments

row sums -Product[3*k - 1, {k, 0, n}]:A008544
{1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400, 2504902400, 72642169600,
2324549427200, 81359229952000, 3091650738176000, 126757680265216000,
5577337931669504000, 262134882788466688000, 13106744139423334400000,
694657439389436723200000,...}

Examples

			{1},
{1, 1},
{1, 8, 1},
{1, 39, 39, 1},
{1, 110, 658, 110, 1},
{1, 1232, 4927, 4927, 1232, 1},
{1, 17453, 34906, 104720, 34906, 17453, 1},
{1, 299200, 598400, 1196799, 1196799, 598400, 299200, 1},
{1, 6021400, 12042800, 18064200, 24085598, 18064200, 12042800, 6021400, 1},
{1, 139161244, 278322488, 417483733, 417483734, 417483734, 417483733, 278322488, 139161244, 1},
{1, 3632108480, 7264216960, 10896325440, 14528433920, -2, 14528433920, 10896325440, 7264216960, 3632108480, 1}

Crossrefs

A008544, A142458, A142459

Programs

  • Mathematica
    Clear[v, n, row, f]; row[n_] = -Product[3*k - 1, {k, 0, n}];
f[n_, m_] = Floor[(m/n)*row[n]/2]; v[0] = {1}; v[1] = {1, 1};
v[n_] := v[n] = If[Mod[n, 2] == 0, Join[{1}, Table[ f[n, m], {m, 1, Floor[ n/2] - 1}], {row[n] - 2*Sum[ f[n, m], {m, 1, Floor[n/2] - 1}] - 2}, Table[ f[n, m], {m, Floor[n/ 2] - 1, 1, -1}], { 1}],
Join[{1}, Table[ f[n, m], {m, 1, Floor[n/2] - 1}], {row[n]/2 - Sum[ f[n, m], { m, 1, Floor[n/2] - 1}] - 1, row[n]/ 2 - Sum[ f[n, m], {m, 1, Floor[ n/2] - 1}] - 1}, Table[ f[n, m], {m, Floor[n/ 2] - 1, 1, -1}], {1}]];
Table[FullSimplify[v[n]], {n, 0, 10}]; Flatten[%]

Formula

row(n)=(2*n)!/n!: f(n,m)=Floor[(m/n)*row(n)].

A155917 A difference triangle of Pascal-Sierpinski 5th level and the Pascal second derivative: a(n,k)= (4*n - 4*k + 1)a(n - 1, k - 1) + (4*k - 3)a(n - 1, k); p(x,n)=(Sum[10*n*(n - 1)*a(n, k)*x^(k - 1) - D[(x + 1)^(n + 2), {x, 2}]/(x + 1), {k, n}])/2.

Original entry on oeis.org

-3, -2, -2, 0, 240, 0, 3360, 3360, -5, 30380, 105570, 30380, -5, -18, 232710, 2032620, 2032620, 232710, -18, -42, 1637748, 31186890, 74043480, 31186890, 1637748, -42, -80, 10932880, 420179760, 1990483600, 1990483600, 420179760, 10932880, -80
Offset: 1

Views

Author

Roger L. Bagula, Jan 30 2009

Keywords

Comments

Row sums are:

Examples

			{-3},
{-2, -2},
{0, 240},
{0, 3360, 3360},
{-5, 30380, 105570, 30380, -5},
{-18, 232710, 2032620, 2032620, 232710, -18},
{-42, 1637748, 31186890, 74043480, 31186890, 1637748, -42},
{-80, 10932880, 420179760, 1990483600, 1990483600, 420179760, 10932880, -80},
{-135, 70305480, 5213648700, 44614752120, 87013084950, 44614752120, 5213648700, 70305480, -135},
{-210, 439442910, 61202397240, 887917071960, 3020166679140, 3020166679140, 887917071960, 61202397240, 439442910, -210}

Crossrefs

A142459

Programs

  • Mathematica
    A[n_, 1] := 1; A[n_, n_] := 1;
A[n_, k_] := (4*n - 4*k + 1)A[n - 1, k - 1] + (4*k - 3)A[n - 1, k];
a = Table[ExpandAll[(Sum[10*n*(n - 1)*A[n, k]*x^(k - 1) - D[(x + 1)^(n + 2), {x, 2}]/(x + 1), {k, n}])/2], {n, 10}];
Table[CoefficientList[ExpandAll[a[[n]]], x], {n, 1, Length[a]}];
Flatten[%]

Formula

a(n,k)= (4*n - 4*k + 1)a(n - 1, k - 1) + (4*k - 3)a(n - 1, k);
p(x,n)=(Sum[10*n*(n - 1)*a(n, k)*x^(k - 1) - D[(x + 1)^(n + 2), {x, 2}]/(x + 1), {k, n}])/2;
t(n,m)=coefficients(p(x,n)).

A157629 A general recursion triangle with third part a power triangle:m=2; Power triangle: f(n,k,m)=If[n*k*(n - k) == 0, 1, n^m - (k^m + (n - k)^m)]; Recursion: A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) + (m*k + 1)*A(n - 1, k, m) + m*f(n, k, m)*A(n - 2, k - 1, m).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 43, 43, 1, 1, 148, 590, 148, 1, 1, 469, 5018, 5018, 469, 1, 1, 1438, 34047, 91492, 34047, 1438, 1, 1, 4351, 204813, 1187731, 1187731, 204813, 4351, 1, 1, 13096, 1149652, 12609880, 27234646, 12609880, 1149652, 13096, 1, 1, 39337
Offset: 0

Views

Author

Roger L. Bagula, Mar 03 2009

Keywords

Comments

Row sums are:
{1, 2, 12, 88, 888, 10976, 162464, 2793792, 54779904, 1206055680, 29460493056,...}.

Examples

			{1},
{1, 1},
{1, 10, 1},
{1, 43, 43, 1},
{1, 148, 590, 148, 1},
{1, 469, 5018, 5018, 469, 1},
{1, 1438, 34047, 91492, 34047, 1438, 1},
{1, 4351, 204813, 1187731, 1187731, 204813, 4351, 1},
{1, 13096, 1149652, 12609880, 27234646, 12609880, 1149652, 13096, 1},
{1, 39337, 6188356, 117961172, 478838974, 478838974, 117961172, 6188356, 39337, 1},
{1, 118066, 32448653, 1015124312, 7053594482, 13257922028, 7053594482, 1015124312, 32448653, 118066, 1}

Crossrefs

A142459, A154336, A157277

Programs

  • Mathematica
    A[n_, 0, m_] := 1; A[n_, n_, m_] := 1;
A[n_, k_, m_] := (m*(n - k) + 1)*A[n - 1, k - 1, m] + (m*k + 1)*A[n - 1, k, m] + m*f[n, k, m]*A[n - 2, k - 1, m];
Table[A[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[A[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[A[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Formula

m=0:Pascal:m=1Eulerian numbers;
m=2;
Power triangle:
f(n,k,m)=If[n*k*(n - k) == 0, 1, n^m - (k^m + (n - k)^m)];
Recursion:
A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) +
(m*k + 1)*A(n - 1, k, m) +
m*f(n, k, m)*A(n - 2, k - 1, m).
Previous Showing 21-25 of 25 results.

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

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