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.

A067302 Row sums of triangle A067298 and of A067304.

Original entry on oeis.org

1, 3, 18, 160, 1752, 21504, 282304, 3867840, 54547200, 785255680, 11478167040, 169748686848, 2533556365312, 38094656593920, 576271774875648, 8761529890717696, 133776598692003840, 2050020136793604096
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Formula

a(n)=sum(A067298(n, m), m=0..n ).
Bisection: a(2*k)= (k+1)*A067297(2*k)=: A067303(k), a(2*k+1)= (2*k+3)*A067297(2*k+1)/2 =: A067322(k), k>=0.
G.f.: ge(x^2) + x*go(x^2) with ge(x) g.f. of A067303 and go(x) g.f. of A067322.

A067329 Triangle of coefficients of polynomials used for g.f.s of columns of A067304.

Original entry on oeis.org

1, 3, 1, 3, 10, 3, 3, 10, 12, 3, 21, 70, 93, 60, 12, 48, 160, 219, 165, 72, 12, 507, 1690, 2343, 1872, 996, 336, 48, 1461, 4870, 6798, 5595, 3252, 1380, 384, 48, 17841, 59470, 83361, 69828, 42636, 20256, 7248, 1728
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Comments

The row polynomials p(n,y) := sum(a(n,m)y^m,m=0..n), n>=1, appear in the g.f.s for the n-th column of triangle A067304.

Examples

			{1}; {3,1}; {3,10,3}; {3,3,10,12}; ...

Formula

a(n, m)=[y^m](p(n, y)), n>=m>=1, a(0, 0)=1, else 0, where p(k, y) is, for k>=1, defined by the g.f. of the k-th column of triangle A067304(n, k).

A067305 Second column of triangle A067304.

Original entry on oeis.org

1, 5, 36, 328, 3440, 39408, 478912, 6068480, 79315200, 1061628160, 14480086016, 200540018688, 2812618092544, 39867889037312, 570237130752000, 8219880968028160, 119293333282291712, 1741605394647416832
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Formula

a(n)= A067304(n+1, 1) = A067297(n+1) - A064340(n+1), n>=0.
G.f.: 4*(3+c(4*x))*(c(4*x)^3)/(1+c(4*x))^4 with c(x) g.f. of A000108 (Catalan).

A067306 One-fourth of third column of triangle A067304.

Original entry on oeis.org

1, 8, 75, 796, 9176, 111936, 1421968, 18618560, 249542400, 3407171584, 47226230528, 662805371904, 9400304896000, 134517761982464, 1939837469085696, 28162286932246528, 411276783645753344
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Formula

a(n)= A067304(n+2, 2)/4 = (A067297(n+2) - (A064340(n+2)+A064340(n+1)))/4, n>=0.
G.f.: (3+10c(4*x)+3*c(4*x)^2)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).

A067307 One-fourth of fourth column of triangle A067304.

Original entry on oeis.org

7, 71, 768, 8920, 109232, 1390800, 18237952, 244701440, 3343713024, 46374830848, 651170275328, 9238908291072, 132251092529152, 1907671386263552, 27701755840561152, 404632598092447744
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Formula

a(n)= A067304(n+3, 3)/4 = (A067297(n+3)-(b(n+3)+b(n+2)+4*b(n+1)))/4, n>=0, with b(n) := A064340(n).
G.f.: 4(3+10*c(4*x)+12*c(4*x)^2+3*c(4*x)^3)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).

A067308 One sixteenth of fifth column of triangle A067304.

Original entry on oeis.org

16, 185, 2181, 26860, 342968, 4504944, 60509296, 827456576, 11482655232, 161302619392, 2289365653760, 32780329073664, 472951175022592, 6869148315201536, 100352220112662528, 1473672361011920896
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Examples

			21+70*y+93*y^2+60*y^3 = p(4,y), fifth row polynomial of triangle A067329.

Formula

a(n)= A067304(n+4, 4)/15 = (A067297(n+4)-sum(b(j)b(n+4-j), j=0..3))/16, n>=0, with b(n) := A064340(n).
G.f.: (21+70c(4*x)+93*c(4*x)^2+60*c(4*x)^3+12*c(4*x)^4)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).

A067298 Generalized Catalan triangle, based on C(2,2; n) = A064340(n).

Original entry on oeis.org

1, 1, 2, 4, 5, 9, 28, 32, 36, 64, 256, 284, 300, 328, 584, 2704, 2960, 3072, 3184, 3440, 6144, 31168, 33872, 34896, 35680, 36704, 39408, 70576, 380608, 411776, 422592, 429760, 436928, 447744, 478912, 859520, 4840960, 5221568, 5346240, 5421952, 5487488, 5563200, 5687872, 6068480, 10909440
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Comments

For corresponding Catalan triangle with C(1,1; n) = A000108(n) see A028364.
Identity for each row n>=1: T(n, m) + T(n, n-(m+1)) = T(n, n) = A067297(n) for m=0..floor((n-1)/2). E.g., T(2*k+1, k) = A067297(2*k+1)/2.

Examples

			Triangle begins:
    1;
    1,   2;
    4,   5,   9;
   28,  32,  36,  64;
  256, 284, 300, 328, 584;
  ...

Crossrefs

The columns (without leading zeros) give for m=0..3: A064340, A067299, 3*A067300, 8*A067301.
The main diagonal gives A067297. The row sums give A067302.
Cf. A000108, A067304.

Programs

  • PARI
    A064340(n) = if(n>1, sum(m=0, n-2, (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)/2^(m+1))*(4^(n-1))/(n-1), 1);
T(n, m) = sum(i=0, m, A064340(i)*A064340(n-i)); \\ Jinyuan Wang, Apr 20 2025

Formula

T(n, m) = Sum_{i=0..m} C(2,2; i)*C(2,2; n-i) if n >= m >= 0 else 0.
G.f. for column m (without leading zeros): (c(m, x)*c(2,2; x)-c2(m-1, x))/x^m, with c(2,2; x) = (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 (g.f. for C(2,2; n)), c(x) = g.f. for Catalan numbers A000108, c(m, x) = Sum_{n=0..m} C(2,2; n)*x^n and c2(m, x) = Sum_{n=0..m} A067297(n)*x^n for m=0, 1, 2, ...

Extensions

More terms from Jinyuan Wang, Apr 20 2025
Showing 1-7 of 7 results.

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