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 11 results. Next

A067320 Even-indexed members of A067297.

Original entry on oeis.org

1, 9, 584, 70576, 10909440, 1913027840, 361936623616, 72033971859456, 14864066521333760, 3151356047929704448, 682330743447507959808, 150243739893975187718144, 33540494675674022306381824
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Crossrefs

Cf. A067321 (one half of odd-indexed terms of A067297).

Formula

a(n) = A067297(2n).
G.f.: A(x) = (f(y) + f(-y))/2 with f(y) := (-1 + 9*y + (1+6*y)*c(y))/((1-y)*(1+5*y)*c(y) - 1 + 11*y - y^2) where c(y) is g.f. of A000108 (Catalan) and y=4*sqrt(x).

A067321 One-half of odd-indexed members of A067297.

Original entry on oeis.org

1, 32, 3072, 429760, 71386880, 13057591296, 2539643772928, 515384111218688, 107895796673347584, 23131454701441122304, 5052953723661587578880, 1120663604347562971103232, 251676943773831910414876672
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Crossrefs

Cf. A067320 (even-indexed members of A067297).

Formula

a(n) = A067297(2n+1)/2.
G.f.: A(x) = (f(y) - f(-y))/y with f(y) := (-1 + 9*y + (1+6*y)*c(y))/((1-y)*(1+5*y)*c(y) - 1 + 11*y - y^2) where c(y) is g.f. of A000108 (Catalan) and y=4*sqrt(x).

A067304 Generalized Catalan triangle A067298 with row reversion.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Comments

Identity for each row n >= 1: T(n, m) + T(n, n-m+1) = A067297(n+1) (convolution of generalized Catalan numbers) for every m = 1..floor((n+1)/2). E.g., T(2*k+1, k+1) = A067297(2*(k+1))/2.

Examples

			Triangle begins:
    1;
    2,   1;
    9,   5,   4;
   64,  36,  32,  28;
  584, 328, 300, 284, 256;
  ...
n=3: T(3, 0) = 64 = 36+28 = 32+32.

Crossrefs

The columns give for m=0..4: A067297 (diagonals of A067298), A067305, A067306, A067307, A067308.
Cf. A067302 (row sums), A067323 (corresponding triangle for ordinary Catalan numbers).
Cf. A000108, A067298, A067329.

Formula

T(n, m) = A067298(n, n-m), n >= m >= 0, otherwise 0.
G.f. for column m >= 1 (without leading zeros): (2^(2*ceiling(m/2))*p(m, y)*(y^3)/(1+y)^4, where y = y(x) = c(4*x), with c(x) = g.f. of A000108 (Catalan) and the row polynomials p(n, y) = Sum_{k=0..n} A067329(n, k)*y^k, n >= 1. For m = 0: ((y*(3+y))^2)/(1+y)^4 with y = y(x) = c(4*x) (see A067297).

Extensions

More terms from Jinyuan Wang, Apr 20 2025

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

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.

A067303 Even-indexed members of A067302.

Original entry on oeis.org

1, 18, 1752, 282304, 54547200, 11478167040, 2533556365312, 576271774875648, 133776598692003840, 31513560479297044480, 7505638177922587557888, 1802924878727702252617728, 436026430783762289982963712
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Crossrefs

Cf. A067322 (odd-indexed members of A067302).

Formula

a(n) = A067302(2n) = (n+1)*A067297(2*n).
G.f.: (x*(d/dx)A(x) + A(x)) with A(x) g.f. of A067320.

A067322 Odd-indexed members of A067302.

Original entry on oeis.org

3, 160, 21504, 3867840, 785255680, 169748686848, 38094656593920, 8761529890717696, 2050020136793604096, 485760548730263568384, 116217935644216514314240, 28016590108689074277580800
Offset: 0

Views

Author

Wolfdieter Lang, Feb 05 2002

Keywords

Crossrefs

Cf. A067303 (even-indexed members of A067302).

Formula

a(n) = A067302(2n+1) = (2*n+3)*A067297(2*n+1)/2.
G.f.: A(x) = (2*x*(d/dx)g(x) + 3*g(x)) with g(x) g.f. of A067321.

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).
Showing 1-10 of 11 results. Next

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

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