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 1-5 of 5 results.

A156644 Mirror image of triangle A080233.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, -2, 0, 2, 1, -3, -2, 2, 3, 1, -4, -5, 0, 5, 4, 1, -5, -9, -5, 5, 9, 5, 1, -6, -14, -14, 0, 14, 14, 6, 1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 12 2009

Keywords

Comments

Inverse of A239473. Equals A007318*A167374. - Tom Copeland, Nov 14 2016

Examples

			Triangle begins as:
   1;
   0,  1;
  -1,  1, 1;
  -2,  0, 2, 1;
  -3, -2, 2, 3, 1;
  -4, -5, 0, 5, 4, 1; ...

Links

Crossrefs

Cf. A024000, A080956, A097808, A120730.
Cf. A007318, A167374, A239473.

Programs

  • Magma
    A156644:= func< n,k | ((2*k-n+1)/(k+1))*Binomial(n,k) >;
[A156644(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2021
  • Mathematica
    Table[Binomial[n, k] -Binomial[n, k+1], {n,0,10}, {k,0,n}]//Flatten (* Michael De Vlieger, Nov 24 2016 *)
  • Sage
    def A156644(n,k): return ((2*k-n+1)/(k+1))*binomial(n,k)
flatten([[A156644(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2021

Formula

T(n,k) = A080233(n,n-k) = (-1)^(n-k)*A097808(n,k).
T(n,k) = ((2*k-n+1)/(k+1))*binomial(n,k).
T(n,k) = T(n-1,k-1) + T(n-1,k), k>0, with T(n,0) = 1-n = A024000(n), T(n,n) = 1.
T(n,k) = binomial(n,k) - binomial(n,k+1) = Sum_{i=-k-1..k+1} (-1)^(i+1) * binomial(n,k+1+i) * binomial(n+2,k+1-i). - Mircea Merca, Apr 28 2012
Sum_{k=0..n} T(n, k) = A000012(n) = 1^n. - G. C. Greubel, Feb 28 2021

A080236 Triangle of differences of consecutive pairs of row elements of triangle A080233.

Original entry on oeis.org

1, 1, -1, 1, 0, -2, 1, 1, -2, -2, 1, 2, -1, -4, -1, 1, 3, 1, -5, -5, 1, 1, 4, 4, -4, -10, -4, 4, 1, 5, 8, 0, -14, -14, 0, 8, 1, 6, 13, 8, -14, -28, -14, 8, 13, 1, 7, 19, 21, -6, -42, -42, -6, 21, 19, 1, 8, 26, 40, 15, -48, -84, -48, 15, 40, 26
Offset: 0

Views

Author

Paul Barry, Feb 09 2003

Keywords

Comments

Row sums are given by -A023443.

Examples

			Rows are {1}, {1,-1}, {1,0,-2}, {1,1,-2,-2}, {1,2,-1,-4,-1}, ...

Crossrefs

Cf. A080233, A007318.

A131428 a(n) = 2*C(n) - 1, where C(n) = A000108(n) are the Catalan numbers.

Original entry on oeis.org

1, 1, 3, 9, 27, 83, 263, 857, 2859, 9723, 33591, 117571, 416023, 1485799, 5348879, 19389689, 70715339, 259289579, 955277399, 3534526379, 13128240839, 48932534039, 182965127279, 686119227299, 2579808294647, 9723892802903, 36734706144303, 139067101832007
Offset: 0

Views

Author

Gary W. Adamson, Jul 10 2007

Keywords

Comments

Starting (1, 3, 9, 27, 83, ...), = row sums of triangle A136522. - Gary W. Adamson, Jan 02 2008
Hankel transform is A171552. - Paul Barry, Dec 11 2009
Apparently, for n >= 1, the maximum peak height minus the maximum valley height summed over all Dyck n-paths (with max valley height deemed zero if no valleys). - David Scambler, Oct 05 2012
Apparently for n > 1 the number of fixed points in all Dyck (n-1)-paths. A fixed point occurs when a vertex of a Dyck k-path is also a vertex of the path U^kD^k. - David Scambler, May 01 2013

Examples

			a(3) = 9 = 2*C(3) - 1 = 2*5 - 1, where C refers to the Catalan numbers, A000108.

Links

Crossrefs

Cf. A000108, A131427, A131429.
Cf. A131428, A118976, A136522.
Cf. A080233, A156644.

Programs

  • GAP
    List([0..25], n-> 2*Binomial(2*n,n)/(n+1) - 1); # G. C. Greubel, Aug 12 2019
  • Magma
    [2*Catalan(n) -1: n in [0..25]]; // G. C. Greubel, Aug 12 2019
  • Maple
    seq(2*binomial(2*n,n)/(n+1)-1, n=0..25); # Emeric Deutsch, Jul 25 2007
  • Mathematica
    2CatalanNumber[Range[0,25]]-1  (* Harvey P. Dale, Apr 17 2011 *)
  • PARI
    vector(25, n, n--; 2*binomial(2*n,n)/(n+1) - 1) \\ G. C. Greubel, Aug 12 2019
  • Sage
    [2*catalan_number(n) -1 for n in (0..25)] # G. C. Greubel, Aug 12 2019

Formula

Right border of triangle A131429.
From Emeric Deutsch, Jul 25 2007: (Start)
a(n) = 2*binomial(2*n,n)/(n+1) - 1.
G.f.: (1-sqrt(1-4*x))/x - 1/(1-x). (End)
(1, 3, 9, 27, 83, ...) = row sums of A118976. - Gary W. Adamson, Aug 31 2007
Row sums of triangle A131428 starting (1, 3, 9, 27, 83, ...). - Gary W. Adamson, Aug 31 2007
Starting with offset 1 = Narayana transform (A001263) of [1,2,2,2,...]. - Gary W. Adamson, Jul 29 2011
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +3*(-n+1)=0. - R. J. Mathar, Nov 22 2024
a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^2 = Sum_{k=0..n} A080233(n,k)^2 = Sum_{k=0..n} A156644(n,k)^2. - Seiichi Manyama, Mar 25 2025

Extensions

More terms from Emeric Deutsch, Jul 25 2007

A382435 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^6.

Original entry on oeis.org

1, 1, 3, 129, 1587, 39443, 1125383, 30211457, 1107074979, 36214609683, 1433494688871, 54495716261011, 2275005440977063, 95146470595975399, 4170974287982618639, 185640304224109725569, 8492643748223480148419, 395051289603660979274339, 18726850582009755291702599
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2025

Keywords

Crossrefs

Cf. A131428, A382434.
Cf. A080233, A156644, A382433.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> 2*add(b(n, n-2*j)^6, j=0..n/2)-1:
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025
  • PARI
    a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^6);
  • Python
    from math import comb
def A382435(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

Formula

a(n) = Sum_{k=0..n} A080233(n,k)^6 = Sum_{k=0..n} A156644(n,k)^6.
a(n) = 2 * A382433(n) - 1.

A382434 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^4.

Original entry on oeis.org

1, 1, 3, 33, 195, 1763, 15623, 156257, 1630947, 17911299, 203739015, 2389928995, 28749060871, 353362388551, 4424242664975, 56290517376737, 726355164976547, 9490129871680355, 125375330053632455, 1672895457018337859, 22522481793315373319, 305695116823973096519
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2025

Keywords

Crossrefs

Cf. A131428, A382435.
Cf. A080233, A129123, A156644.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> 2*add(b(n, n-2*j)^4, j=0..n/2)-1:
seq(a(n), n=0..21);  # Alois P. Heinz, Mar 25 2025
  • PARI
    a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^4);
  • Python
    from math import comb
def A382434(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**4 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

Formula

a(n) = Sum_{k=0..n} A080233(n,k)^4 = Sum_{k=0..n} A156644(n,k)^4.
a(n) = 2 * A129123(n) - 1.
D-finite with recurrence n*(n+1)^3*a(n) -2*n*(11*n^3-17*n^2+5*n+5)*a(n-1) -4*(n-1)*(70*n^3-365*n^2+527*n-162)*a(n-2) +8*(n-2)*(584*n^3-5020*n^2+14111*n-13059)*a(n-3) +1344*(4*n-11)*(4*n-13)*(-3+n)^2*a(n-4) +9*(2875*n^4-33975*n^3+149945*n^2-293541*n+215336)=0. - R. J. Mathar, Mar 31 2025
Showing 1-5 of 5 results.

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