A081583 -id:A081583 - cp's OEIS frontend

A081577 Pascal-(1,2,1) array read by antidiagonals.

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 22, 10, 1, 1, 13, 46, 46, 13, 1, 1, 16, 79, 136, 79, 16, 1, 1, 19, 121, 307, 307, 121, 19, 1, 1, 22, 172, 586, 886, 586, 172, 22, 1, 1, 25, 232, 1000, 2086, 2086, 1000, 232, 25, 1, 1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016777, A038764, A081583, A081584. Coefficients of the row polynomials in the Newton basis are given by A013610.
As a number triangle, this is the Riordan array (1/(1-x), x(1+2x)/(1-x)). It has row sums A002605 and diagonal sums A077947. - Paul Barry, Jan 24 2005
All entries are == 1 mod 3. - Roger L. Bagula, Oct 04 2008
Row sums are A002605. - Roger L. Bagula, Dec 09 2008
As a number triangle T, T(2n,n)=A069835(n). - Philippe Deléham, Jan 10 2014

			Square array begins as:
  1,  1,  1,   1,   1, ... A000012;
  1,  4,  7,  10,  13, ... A016777;
  1,  7, 22,  46,  79, ... A038764;
  1, 10, 46, 136, 307, ... A081583;
  1, 13, 79, 307, 886, ... A081584;
From _Roger L. Bagula_, Dec 09 2008: (Start)
As a triangle this begins:
  1;
  1,  1;
  1,  4,   1;
  1,  7,   7,    1;
  1, 10,  22,   10,    1;
  1, 13,  46,   46,   13,    1;
  1, 16,  79,  136,   79,   16,    1;
  1, 19, 121,  307,  307,  121,   19,    1;
  1, 22, 172,  586,  886,  586,  172,   22,   1;
  1, 25, 232, 1000, 2086, 2086, 1000,  232,  25,  1;
  1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1; (End)

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Bala, A note on the diagonals of a proper Riordan Array
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Shanghua Zheng, Li Guo, and Huizhen Qiu, Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras, arXiv:2401.11363 [math.RA], 2024. See pp. 44-45.

Cf. A002605, A081578, A081579, A081580.

Cf. Pascal-(1,a,1) array: A123562 (a=-3), A098593 (=-2), A000012 (a=-1), A007318 (a=0), A008288 (a=1), A081577(a=2), A081578 (a=3), A081579 (a=4), A081580 (a=5), A081581 (a=6), A081582 (a=7), A143683(a=8). [From Roger L. Bagula, Dec 09 2008], Philippe Deléham, Jan 10 2014, Mar 16 2014.

a081577 n k = a081577_tabl !! n !! k
a081577_row n = a081577_tabl !! n
a081577_tabl = map fst $ iterate
    (\(us, vs) -> (vs, zipWith (+) (map (* 2) ([0] ++ us ++ [0])) $
                       zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A081577:= func< n,k | (&+[Binomial(k,j)*Binomial(n-j,k)*2^j: j in [0..n-k]]) >;
[A081577(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 25 2021

a[0]={1}; a[1]={1, 1}; a[n_]:= a[n]= 2*Join[{0}, a[n-2], {0}] + Join[{0}, a[n-1]] + Join[a[n-1], {0}]; Table[a[n], {n,0,10}]//Flatten (* Roger L. Bagula, Dec 09 2008 *)
Table[Hypergeometric2F1[-k, k-n, 1, 3], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 3).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 2*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+2*x)^k/(1-x)^(k+1).
G.f.: 1/(1-x-y-2*x*y). - Ralf Stephan, Apr 28 2004
T(n,k) = Sum_{j=0..n} binomial(k,j-k)*binomial(n+k-j,k)*2^(j-k). - Paul Barry, Oct 23 2006
a(n) = 2*{0, a(n-2), 0} + {0, a(n-1)} + {a(n-1), 0}. - Roger L. Bagula, Dec 09 2008
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 3). - Jean-François Alcover, May 24 2013
The 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)*(3*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 6*x + 9*x^2/2) = 1 + 7*x + 22*x^2/2! + 46*x^3/3! + 79*x^4/4! + 121*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A002605(n). - G. C. Greubel, May 25 2021

A081584 Fourth row of Pascal-(1,2,1) array A081577.

Original entry on oeis.org

1, 13, 79, 307, 886, 2086, 4258, 7834, 13327, 21331, 32521, 47653, 67564, 93172, 125476, 165556, 214573, 273769, 344467, 428071, 526066, 640018, 771574, 922462, 1094491, 1289551, 1509613, 1756729, 2033032, 2340736, 2682136, 3059608

Offset: 0

Paul Barry, Mar 23 2003

Equals binomial transform of [1, 12, 54, 108, 81, 0, 0, 0, ...] where (1, 12, 54, 108, 81) = row 4 of triangle A013610. - Gary W. Adamson, Jul 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A013610, A081583.

[(8+6*n+81*n^2-18*n^3+27*n^4)/8: n in [0..40]]; // Vincenzo Librandi, Aug 09 2013

seq((8+6*n+81*n^2-18*n^3+27*n^4)/8, n=0..40); # G. C. Greubel, May 26 2021

CoefficientList[Series[(1+2x)^4/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,13,79,307,886},40] (* Harvey P. Dale, Sep 18 2024 *)

[(8+6*n+81*n^2-18*n^3+27*n^4)/8 for n in (0..40)] # G. C. Greubel, May 26 2021

a(n) = (8 + 6*n + 81*n^2 - 18*n^3 + 27*n^4)/8.
G.f.: (1+2*x)^4/(1-x)^5.
E.g.f.: (1/8)*(8 + 96*x + 216*x^2 + 144*x^3 + 27*x^4)*exp(x). - G. C. Greubel, May 26 2021

A361731 Array read by descending antidiagonals. A(n, k) = hypergeom([-k, -3], [1], n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 7, 1, 1, 20, 25, 10, 1, 1, 35, 63, 46, 13, 1, 1, 56, 129, 136, 73, 16, 1, 1, 84, 231, 307, 245, 106, 19, 1, 1, 120, 377, 586, 593, 396, 145, 22, 1, 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1, 1, 220, 833, 1576, 2073, 2076, 1585, 848, 241, 28, 1

Offset: 0

Peter Luschny, Mar 22 2023

			Array A(n, k) starts:
 [0] 1,  1,   1,   1,    1,    1,    1,     1, ...  A000012
 [1] 1,  4,  10,  20,   35,   56,   84,   120, ...  A000292
 [2] 1,  7,  25,  63,  129,  231,  377,   575, ...  A001845
 [3] 1, 10,  46, 136,  307,  586, 1000,  1576, ...  A081583
 [4] 1, 13,  73, 245,  593, 1181, 2073,  3333, ...  A081586
 [5] 1, 16, 106, 396, 1011, 2076, 3716,  6056, ...  A081588
 [6] 1, 19, 145, 595, 1585, 3331, 6049,  9955, ...  A081590
 [7] 1, 22, 190, 848, 2339, 5006, 9192, 15240, ...
.
Table T(n, k) starts:
 [0] 1;
 [1] 1,   1;
 [2] 1,   4,   1;
 [3] 1,  10,   7,    1;
 [4] 1,  20,  25,   10,    1;
 [5] 1,  35,  63,   46,   13,    1;
 [6] 1,  56, 129,  136,   73,   16,   1;
 [7] 1,  84, 231,  307,  245,  106,  19,   1;
 [8] 1, 120, 377,  586,  593,  396, 145,  22,  1;
 [9] 1, 165, 575, 1000, 1181, 1011, 595, 190, 25, 1;

Rows: A000012, A000292, A001845, A081583, A081586, A081588, A081590.
Columns: A000012, A016777, A100536.
Hypergeometric family: A000012 (m=0), A077028 (m=1), A361682 (m=2), this array (m=3).

A := (n, k) -> 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6;
seq(print(seq(A(n, k), k = 0..7)), n = 0..7);
# Alternative:
ogf := n -> (1 + (n - 1) * x)^3 / (1 - x)^4:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..9);

A(n, k) = [x^k] (1 + (n - 1) * x)^3 / (1 - x)^4.
A(n, k) = 1 + (((k*n - 3*n + 9)*n*k + (2*n - 9)*n + 18)*n*k)/6.
T(n, k) = 1 + (((k*(n - k) - 3*k + 9)*k*(n - k) + (2*k - 9)*k + 18)*k*(n - k))/6.

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A081577 Pascal-(1,2,1) array read by antidiagonals.

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A081584 Fourth row of Pascal-(1,2,1) array A081577.

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A361731 Array read by descending antidiagonals. A(n, k) = hypergeom([-k, -3], [1], n).

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