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 1-7 of 7 results.

A176700 Triangle T(n,m) = 2+A176697(n)-A176697(m)-A176697(n-m) read along rows 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 14, 16, 14, 1, 1, 44, 55, 55, 44, 1, 1, 146, 187, 196, 187, 146, 1, 1, 504, 647, 686, 686, 647, 504, 1, 1, 1786, 2287, 2428, 2458, 2428, 2287, 1786, 1, 1, 6449, 8232, 8731, 8863, 8863, 8731, 8232, 6449, 1, 1, 23635, 30081, 31862, 32352
Offset: 0

Views

Author

Roger L. Bagula, Apr 24 2010

Keywords

Comments

Row sums are 1, 2, 5, 12, 46, 200, 864, 3676, 15462, 64552, 268316,...

Examples

			1;
1, 1;
1, 3, 1;
1, 5, 5, 1;
1, 14, 16, 14, 1;
1, 44, 55, 55, 44, 1;
1, 146, 187, 196, 187, 146, 1;
1, 504, 647, 686, 686, 647, 504, 1;
1, 1786, 2287, 2428, 2458, 2428, 2287, 1786, 1;
1, 6449, 8232, 8731, 8863, 8863, 8731, 8232, 6449, 1;
1, 23635, 30081, 31862, 32352, 32454, 32352, 31862, 30081, 23635, 1;

Crossrefs

Cf. A176697.

Programs

  • Maple
    A176700 :=proc(n,k)
    2+A176697(n)-A176697(k)-A176697(n-k) ;
end proc: # R. J. Mathar, Jun 17 2015
  • Mathematica
    a[0] := 1; a[1] := 1;a[2]=3
a[n_] := a[n] = Table[a[i], {i, 0, n - 1}].Table[a[n - 1 - i], {i, 0, n - 1}];
t[n_, m_] := 2 + (-a[m] - a[n - m] + a[n]);
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]

Formula

T(n,k) = T(n,n-k).

A366676 G.f. satisfies A(x) = 1 + x^3 + x*A(x)^3.

Original entry on oeis.org

1, 1, 3, 13, 58, 288, 1512, 8250, 46296, 265491, 1548976, 9165156, 54865737, 331694167, 2022232068, 12419023617, 76755164643, 477049187268, 2979758649996, 18695276174079, 117766227611046, 744527923478730, 4722464911515423, 30044091589750350
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2023

Keywords

Crossrefs

Cf. A176697, A366677.
Cf. A366266, A366555.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(2*(n-3*k)+1, k)*binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*(n-3*k)+1,k) * binomial(3*(n-3*k),n-3*k)/(2*(n-3*k)+1).

A367041 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^4.

Original entry on oeis.org

1, 1, 5, 26, 168, 1195, 8988, 70318, 566388, 4665221, 39113732, 332691758, 2863778072, 24900264326, 218372530380, 1929363592870, 17157018725000, 153442147343648, 1379250344938676, 12453816724761706, 112907775890596400, 1027394297869071687
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A176697, A367040.
Cf. A365178, A365180, A365181, A365182, A365183, A367048, A367049, A367050.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(3*(n-2*k)+1, k)*binomial(4*(n-2*k), n-2*k)/(3*(n-2*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*(n-2*k)+1,k) * binomial(4*(n-2*k),n-2*k)/(3*(n-2*k)+1).

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(2*(n-2*k)+1, k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*(n-2*k)+1,k) * binomial(3*(n-2*k),n-2*k)/(2*(n-2*k)+1).

A367042 G.f. satisfies A(x) = 1 + x^3 + x*A(x)^2.

Original entry on oeis.org

1, 1, 2, 6, 16, 48, 152, 500, 1688, 5816, 20368, 72288, 259424, 939808, 3432192, 12622416, 46706144, 173762016, 649569216, 2438748864, 9191656192, 34765298944, 131912452864, 501987944832, 1915417307392, 7326620001536, 28088736525824, 107913607531520
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A366676, A367043.
Cf. A025227, A176697.
Cf. A226022.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-3*k+1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));

Formula

G.f.: A(x) = 2*(1+x^3) / (1+sqrt(1-4*x*(1+x^3))).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-3*k+1,k) * binomial(2*(n-3*k),n-3*k)/(n-3*k+1).

A366677 G.f. satisfies A(x) = 1 + x^4 + x*A(x)^4.

Original entry on oeis.org

1, 1, 4, 22, 141, 973, 7112, 54040, 422552, 3377770, 27478568, 226753828, 1893462584, 15969598554, 135842638632, 1164075017512, 10039732285528, 87081507756245, 759128176746864, 6647475055207618, 58445784269830824, 515745587816906733
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2023

Keywords

Crossrefs

Cf. A176697, A366676.
Cf. A366267, A366558.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(3*(n-4*k)+1, k)*binomial(4*(n-4*k), n-4*k)/(3*(n-4*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(3*(n-4*k)+1,k) * binomial(4*(n-4*k),n-4*k)/(3*(n-4*k)+1).

A176701 A polynomial coefficient triangle sequence:a(n)=vector(a(n-1)).Reverse(vector(a(n-1));a(0)=1;a(1)=1;a[2]=3;p(x,n)=Sum[a(m)*m!*Binomial[x, m], {m, 0, n}].

Original entry on oeis.org

1, 1, 1, 1, -2, 3, 1, 12, -18, 7, 1, -108, 202, -113, 20, 1, 1404, -2948, 2092, -610, 63, 1, -23556, 54044, -44708, 17070, -3057, 208, 1, 488364, -1200160, 1109956, -505515, 121368, -14723, 711, 1, -12091476, 31417568, -31667516, 16389909, -4770792
Offset: 0

Views

Author

Roger L. Bagula, Apr 24 2010

Keywords

Comments

Row sums are:
{1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,...}.

Examples

			{1},
{1, 1},
{1, -2, 3},
{1, 12, -18, 7},
{1, -108, 202, -113, 20},
{1, 1404, -2948, 2092, -610, 63},
{1, -23556, 54044, -44708, 17070, -3057, 208},
{1, 488364, -1200160, 1109956, -505515, 121368, -14723, 711},
{ 1, -12091476, 31417568, -31667516, 16389909, -4770792, 788989, -69177, 2496},
{1, 348530604, -948701728, 1024833540, -585398187, 196013064, -39780995, 4814247, -319488, 8944},
{1, -11473374036, 32495091200, -37179387060, 22990648853, -8578056786, 2021526799, -303047853, 28023372, -1457066, 32578}

Crossrefs

Cf. A176697

Programs

  • Mathematica
    a[0] := 1; a[1] := 1;a[2]=3
a[n_] := a[n] = Table[a[i], {i, 0, n - 1}].Table[a[n - 1 - i], {i, 0, n - 1}];
p[x_, n_] := Sum[a[m]*m!*Binomial[x, m], {m, 0, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

Formula

a(n)=vector(a(n-1)).Reverse(vector(a(n-1));
a(0)=1;a(1)=1;a[2]=3;
p(x,n)=Sum[a(m)*m!*Binomial[x, m], {m, 0, n}];
t(n,m)=coefficients(p(x,n))
Showing 1-7 of 7 results.

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