A286033 -id:A286033 - cp's OEIS frontend

A028331 Elements to the right of the central elements of the even-Pascal triangle A028326 that are not 2.

6, 8, 20, 10, 30, 12, 70, 42, 14, 112, 56, 16, 252, 168, 72, 18, 420, 240, 90, 20, 924, 660, 330, 110, 22, 1584, 990, 440, 132, 24, 3432, 2574, 1430, 572, 156, 26, 6006, 4004, 2002, 728, 182, 28, 12870, 10010, 6006, 2730, 910, 210, 30, 22880, 16016

Offset: 0

Mohammad K. Azarian

			This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
  x;
  .,  .;
  .,  x,  .;
  .,  .,  6,  .;
  .,  .,  x,  8,  .;
  .,  .,  ., 20, 10,   .;
  .,  .,  .,  x, 30,  12,   .;
  .,  .,  .,  ., 70,  42,  14,    .;
  .,  .,  .,  .,  x, 112,  56,   16,   .;
  .,  .,  .,  .,  ., 252, 168,   72,  18,   .;
  .,  .,  .,  .,  .,   x, 420,  240,  90,  20,   .;
  .,  .,  .,  .,  .,   ., 924,  660, 330, 110,  22,  .;
  .,  .,  .,  .,  .,   .,   x, 1584, 990, 440, 132, 24, .;
As an irregular triangle:
    6;
    8;
   20,  10;
   30,  12;
   70,  42,  14;
  112,  56,  16;
  252, 168,  72,  18;
  420, 240,  90,  20;
  924, 660, 330, 110, 22;

G. C. Greubel, Rows n = 0..100 of the irregular triangle, flattened

Cf. A028326, A028327, A028328, A028329, A028330, A028332.

Cf. A272514, A286033.

[2*Binomial(n+3,k): k in [Floor((n+5)/2)..n+2], n in [0..12]]; // G. C. Greubel, Jul 14 2024

Table[2*Binomial[n+3, k+2 +Floor[(n+1)/2]], {n,0,12}, {k,0,Floor[n/2] }]//Flatten (* G. C. Greubel, Jul 14 2024 *)

def A028326(n,k): return 2*binomial(n, k)
flatten([[A028326(n+1,k) for k in range(((n+3)//2), n+1)] for n in range(21)]) # G. C. Greubel, Jul 14 2024

From G. C. Greubel, Jul 14 2024: (Start)
T(n, k) = 2*binomial(n+3, k+2 + floor((n+1)/2)).
Sum_{k=0..floor(n/2)} T(n, k) = A272514(n+3).
Sum_{k=0..n} (-1)^k*T(2*n, k) = 2*A286033(n+2).
Sum_{k=0..n} (-1)^k*T(2*n+1, k) = binomial(2*n+4, n+2) + 2*(-1)^n.
(End)

More terms from James Sellers

A124051 Quasi-mirror of A062196 formatted as a triangular array.

Original entry on oeis.org

3, 6, 8, 10, 30, 15, 15, 80, 90, 24, 21, 175, 350, 210, 35, 28, 336, 1050, 1120, 420, 48, 36, 588, 2646, 4410, 2940, 756, 63, 45, 960, 5880, 14112, 14700, 6720, 1260, 80, 55, 1485, 11880, 38808, 58212, 41580, 13860, 1980, 99, 66, 2200, 22275, 95040, 194040, 199584, 103950, 26400, 2970, 120

Offset: 0

Zerinvary Lajos, Nov 03 2006

			Triangle begins as:
   3;
   6,    8;
  10,   30,    15;
  15,   80,    90,    24;
  21,  175,   350,   210,     35;
  28,  336,  1050,  1120,    420,     48;
  36,  588,  2646,  4410,   2940,    756,     63;
  45,  960,  5880, 14112,  14700,   6720,   1260,    80;
  55, 1485, 11880, 38808,  58212,  41580,  13860,  1980,   99;
  66, 2200, 22275, 95040, 194040, 199584, 103950, 26400, 2970, 120;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A000894, A062196, A286033.
Columns k: A000217(n+2) (k=0), A002417(n+1) (k=1), A001297(n) (k=2), A105946(n-2) (k=3), A105947(n-3) (k=4), A105948(n-4) (k=5), A107319(n-5) (k=6).
Diagonals: A005563(n+1) (k=n), A033487(n) (k=n-1), A027790(n) (k=n-2), A107395(n-3) (k=n-3), A107396(n-4) (k=n-4), A107397(n-5) (k=n-5), A107398(n-6) (k=n-6), A107399(n-7) (k=n-7).
Sums: A322938(n+1) (row).

A124051:= func< n,k | Binomial(n+1,n-k+1)*Binomial(n+3,n-k+1) >;
[A124051(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2025

for n from 0 to 10 do seq(binomial(n,i-1)*binomial(n+2,n+1-i), i=1..n ) od;

A124051[n_, k_]:= Binomial[n+1,n-k+1]*Binomial[n+3,n-k+1];
Table[A124051[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2025 *)

def A124051(n,k): return binomial(n+1,n-k+1)*binomial(n+3,n-k+1)
print(flatten([[A124051(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 07 2025

From G. C. Greubel, Feb 07 2025: (Start)
T(n, k) = binomial(n+1, n-k+1)*binomial(n+3, n-k+1).
T(2*n, n) = (1/2)*A000894(n) + (5/2)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n)*(-1)^(n/2)*A286033((n+4)/2) + (1-(-1)^n)*((-1)^((n+1)/2)*A000108((n+1)/2) - 1) ). (End)

cp's OEIS Frontend

A028331 Elements to the right of the central elements of the even-Pascal triangle A028326 that are not 2.

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