A081582 -id:A081582 - cp's OEIS frontend

A143683 Pascal-(1,8,1) array.

1, 1, 1, 1, 10, 1, 1, 19, 19, 1, 1, 28, 118, 28, 1, 1, 37, 298, 298, 37, 1, 1, 46, 559, 1540, 559, 46, 1, 1, 55, 901, 4483, 4483, 901, 55, 1, 1, 64, 1324, 9856, 21286, 9856, 1324, 64, 1, 1, 73, 1828, 18388, 67006, 67006, 18388, 1828, 73, 1, 1, 82, 2413, 30808, 164242, 304300, 164242, 30808, 2413, 82, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 10,   19,    28,     37,      46,       55, ... A017173;
  1, 19,  118,   298,    559,     901,     1324, ...
  1, 28,  298,  1540,   4483,    9856,    18388, ...
  1, 37,  559,  4483,  21286,   67006,   164242, ...
  1, 46,  901,  9856,  67006,  304300,  1004590, ...
  1, 55, 1324, 18388, 164242, 1004590,  4443580, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 10,   1;
  1, 19,  19,    1;
  1, 28, 118,   28,    1;
  1, 37, 298,  298,   37,   1;
  1, 46, 559, 1540,  559,  46,  1;
  1, 55, 901, 4483, 4483, 901, 55, 1;

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Cf.Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7).

Cf. A003683, A017173, A143680.

a143683 n k = a143683_tabl !! n !! k
a143683_row n = a143683_tabl !! n
a143683_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) (map (* 8) ([0] ++ us ++ [0])) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A143683:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143683(n,k,8): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

Table[Hypergeometric2F1[-k, k-n, 1, 9], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 9).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Square array: T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 8*T(n-1, k-1) + T(n-1, k).
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*9^j.
Rows are the expansions of (1+8*x)^k/(1-x)^(k+1).
Riordan array (1/(1-x), x*(1+8*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 9). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(9*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 18*x + 81*x^2/2) = 1 + 19*x + 118*x^2/2! + 298*x^3/3! + 559*x^4/4! + 901*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A003683(n+1). - G. C. Greubel, May 27 2021

A143685 Pascal-(1,9,1) array.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 141, 31, 1, 1, 41, 361, 361, 41, 1, 1, 51, 681, 1991, 681, 51, 1, 1, 61, 1101, 5921, 5921, 1101, 61, 1, 1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1, 1, 81, 2241, 24681, 96201, 96201, 24681, 2241, 81, 1, 1, 91, 2961, 41511, 239241, 460251, 239241, 41511, 2961, 91, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 11,   21,    31,     41,      51,       61, ... A017281;
  1, 21,  141,   361,    681,    1101,     1621, ...
  1, 31,  361,  1991,   5921,   13151,    24681, ...
  1, 41,  681,  5921,  29761,   96201,   239241, ...
  1, 51, 1101, 13151,  96201,  460251,  1565301, ...
  1, 61, 1621, 24681, 239241, 1565301,  7272861, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 11,    1;
  1, 21,   21,     1;
  1, 31,  141,    31,     1;
  1, 41,  361,   361,    41,     1;
  1, 51,  681,  1991,   681,    51,    1;
  1, 61, 1101,  5921,  5921,  1101,   61,  1;
  1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Cf. A002534, A143680, A143682.

Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8), this sequence (m = 9).

A143685:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143685(n,k,9): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021

Table[Hypergeometric2F1[-k, k-n, 1, 10], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 10).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021

Square array: T(n, k) = T(n, k-1) + 9*T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(0, k) = 1.
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*10^j.
Riordan array (1/(1-x), x*(1+9*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 10). - Jean-François Alcover, May 24 2013
Sum_{k=0..n} T(n, k) = A002534(n+1). - G. C. Greubel, May 29 2021

A174346 Triangle T(n, k) = (binomial(n-1, k-1)binomial(n, k-1)/k) ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 18, 18, 1, 1, 30, 180, 30, 1, 1, 45, 450, 450, 45, 1, 1, 63, 945, 4725, 945, 63, 1, 1, 84, 1764, 13230, 13230, 1764, 84, 1, 1, 108, 3024, 31752, 142884, 31752, 3024, 108, 1, 1, 135, 4860, 68040, 428652, 428652, 68040, 4860, 135, 1

Offset: 1

Roger L. Bagula, Mar 16 2010

			Triangle begins as:
  1;
  1,   1;
  1,   9,    1;
  1,  18,   18,     1;
  1,  30,  180,    30,      1;
  1,  45,  450,   450,     45,      1;
  1,  63,  945,  4725,    945,     63,     1;
  1,  84, 1764, 13230,  13230,   1764,    84,    1;
  1, 108, 3024, 31752, 142884,  31752,  3024,  108,   1;
  1, 135, 4860, 68040, 428652, 428652, 68040, 4860, 135, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A081582.

function T(n,k)
  if Floor(n/2) gt k-1 then return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(k-1);
  else return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(n-k);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 26 2021

T[n_,k_]:= (Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>k-1, 3^(k-1), 3^(n-k)];
Table[T[n,k], {n,12}, {k,n}]//Flatten

def A174346(n,k): return (1/n)*binomial(n,k)*binomial(n,k-1)*( 3^(k-1) if ((n//2)>k-1) else 3^(n-k) )
flatten([[A174346(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Nov 26 2021

T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ).

T(n, n-k) = T(n, k).

Edited by G. C. Greubel, Nov 26 2021

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A143683 Pascal-(1,8,1) array.

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A143685 Pascal-(1,9,1) array.

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A174346 Triangle T(n, k) = (binomial(n-1, k-1)binomial(n, k-1)/k) ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.

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cp's OEIS Frontend

A143683 Pascal-(1,8,1) array.

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A143685 Pascal-(1,9,1) array.

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Magma

Mathematica

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Formula

A174346 Triangle T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.

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A174346 Triangle T(n, k) = (binomial(n-1, k-1)binomial(n, k-1)/k) ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.