A382433 -id:A382433 - cp's OEIS frontend

A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 6, 3, 1, 1, 2, 9, 14, 10, 4, 1, 1, 2, 17, 36, 42, 20, 4, 1, 1, 2, 33, 98, 190, 132, 35, 5, 1, 1, 2, 65, 276, 882, 980, 429, 70, 5, 1, 1, 2, 129, 794, 4150, 7812, 5705, 1430, 126, 6, 1, 1, 2, 257, 2316, 19722, 65300, 78129, 33040, 4862, 252, 6

Offset: 0

Alois P. Heinz, Oct 14 2022

			Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,        1,         1, ...
  1,  1,   1,    1,     1,       1,        1,         1, ...
  2,  2,   2,    2,     2,       2,        2,         2, ...
  2,  3,   5,    9,    17,      33,       65,       129, ...
  3,  6,  14,   36,    98,     276,      794,      2316, ...
  3, 10,  42,  190,   882,    4150,    19722,     94510, ...
  4, 20, 132,  980,  7812,   65300,   562692,   4939220, ...
  4, 35, 429, 5705, 78129, 1083425, 15105729, 211106945, ...

Alois P. Heinz, Antidiagonals n = 0..121, flattened
Wikipedia, Counting lattice paths

Columns k=0-7 give A008619, A001405, A000108, A003161, A129123, A361887, A382433, A361890.
Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.
Main diagonal gives A357825.
Cf. A008315, A120730.

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, k)-> add(b(n, n-2*j)^k, j=0..n/2):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y + j], {j, {-1, 1}}]]];
A[n_, k_] := Sum[b[n, n - 2*j]^k, { j, 0, n/2}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 18 2022, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..floor(n/2)} A008315(n,j)^k.

A(n,k) = Sum_{j=0..n} A120730(n,j)^k for k>=1, A(n,0) = A008619(n).

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

Seiichi Manyama, Mar 25 2025

nonn

Cf. A131428, A382434.

Cf. A080233, A156644, A382433.

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

a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^6);

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

a(n) = Sum_{k=0..n} A080233(n,k)^6 = Sum_{k=0..n} A156644(n,k)^6.

a(n) = 2 * A382433(n) - 1.

A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

Original entry on oeis.org

1, 1, 4, 17, 86, 472, 2752, 16753, 105394, 680366, 4484360, 30067160, 204508240, 1408057120, 9796738304, 68786005361, 486845236106, 3470187822754, 24891491746792, 179556655434382, 1301857088258836, 9482632068303296, 69361538748381824, 509303099950899352

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A129123, A382433, A382443, A382446.
Cf. A131428, A178824.
Cf. A000172, A086618, A238115.

[ &+[Binomial(n, k)^2 * (Binomial(n, k) - (k gt 0 select Binomial(n, k-1) else 0)) : k in [0..n]] : n in [0..20] ]; // Vincenzo Librandi, Mar 27 2025

Table[Sum[Binomial[n,k]^2*(Binomial[n,k]-Binomial[n,k-1]),{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 27 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^2);

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) ).

a(n) ~ 2^(3*n+3) / (Pi * 3^(3/2) * n^2). - Vaclav Kotesovec, Mar 26 2025

A382443 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^4.

Original entry on oeis.org

1, 1, 4, 65, 566, 10912, 164032, 3237313, 62253130, 1314421886, 28392213224, 639799858304, 14785604868256, 350615631856960, 8485316740880384, 209179475361783233, 5239271305444731698, 133100429387161703962, 3424142506153260211720, 89090362800169426107070

Offset: 0

Seiichi Manyama, Mar 26 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A000108, A129123, A381676, A382433, A382446.

Cf. A382434.

[&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^4: k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Mar 29 2025

Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^4,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 29 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^4);

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

a(n) ~ 3 * 2^(5*n+6) / (Pi^2 * 5^(5/2) * n^4). - Vaclav Kotesovec, Mar 26 2025

A382446 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^6.

Original entry on oeis.org

1, 1, 4, 257, 4286, 258952, 11816512, 632854273, 43732565914, 2637804065366, 207379028199080, 14568483339859880, 1205457271871693920, 95108827011788280160, 8187664948710535579904, 698818327346476962092801, 62477582066507173352034866, 5627626080883126186936773514

Offset: 0

Seiichi Manyama, Mar 26 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000108, A129123, A381676, A382433, A382443.

Cf. A382435.

[&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^6: k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Mar 30 2025

Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^6,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 30 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^6);

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^5.

a(n) ~ 15 * 2^(7*n+9) / (Pi^3 * 7^(7/2) * n^6). - Vaclav Kotesovec, Mar 26 2025

cp's OEIS Frontend

A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

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A382443 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^4.

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A382446 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^6.

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