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A172431 Even row Pascal-square read by antidiagonals. 7
1, 1, 2, 1, 4, 3, 1, 6, 10, 4, 1, 8, 21, 20, 5, 1, 10, 36, 56, 35, 6, 1, 12, 55, 120, 126, 56, 7, 1, 14, 78, 220, 330, 252, 84, 8, 1, 16, 105, 364, 715, 792, 462, 120, 9, 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Apart from signs identical to A053123. Mirror of A078812.

As a triangle, row n consists of the coefficients of Morgan-Voyce polynomial B(n,x); e.g., B(3,x)=x^3+6x^2+10x+4. As a triangle, rows 0 to 4 are as follows: 1 1...2 1...4...3 1...6...10...4 1...8...21...20...5 See A054142 for coefficients of Morgan-Voyce polynomial b(n,x).

Scaled version of A119900. - Philippe Deléham, Feb 24 2012

A172431 is jointly generated with A054142 as an array of coefficients of polynomials v(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 09 2012

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012

LINKS

G. C. Greubel, Rows n = 1..100 of triangle, flattened

FORMULA

As a decimal sequence: a(n)= 12*a(n-1)- a(n-2) with a(1)=1. [I interpret this remark as: 1, 12=1,2, 143=1,4,3, 1704=1,6,10,4,... taken from A004191 are decimals on the diagonal. - R. J. Mathar, Sep 08 2013]

As triangle T(n,k): T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012

As DELTA-triangle T(n,k) with 0<=k<=n: G.f.: (1-y*x)^2/((1-y*x)^2-x). - Philippe Deléham, Mar 22 2012

T(n, k) = GegenbauerC(k, n-k, 1). - Peter Luschny, May 10 2016

As triangle T(n,k): Product_{k=1..n} T(n,k) = Product_{k=0..n-1} binomial(2*k,k) = A007685(n-1)  for n >= 1. - Werner Schulte, Apr 26 2017

As triangle T(n,k) with 1 <= k <= n: T(n,k) = binomial(2*n-k, k-1). - Paul Weisenhorn, Nov 25 2019

EXAMPLE

Array begins:

  1,  2,  3,  4,  5,  6, ...

  1,  4, 10, 20, 35, ...

  1,  6, 21, 56, ...

  1,  8, 36, ...

  1, 10, ...

  1, ...

  ...

Example:

Starting with 1, every entry is twice the one to the left minus the second one to the left, plus the one above.

For n = 9 the a(9)= 10 solution is 2*4 - 1 + 3.

From Philippe Deléham, Feb 24 2012: (Start)

Triangle T(n,k) begins:

  1;

  1,   2;

  1,   4,   3;

  1,   6,  10,   4;

  1,   8,  21,  20,   5;

  1,  10,  36,  56,  35,   6;

  1,  12,  55, 120, 126,  56,   7; (End)

From Philippe Deléham, Mar 22 2012: (Start)

(1, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, ...) begins:

  1;

  1,   0;

  1,   2,   0;

  1,   4,   3,   0;

  1,   6,  10,   4,   0;

  1,   8,  21,  20,   5,   0;

  1,  10,  36,  56,  35,   6,   0;

  1,  12,  55, 120, 126,  56,   7,   0; (End)

MAPLE

T := (n, k) -> simplify(GegenbauerC(k, n-k, 1)):

for n from 0 to 10 do seq(T(n, k), k=0..n-1) od; # Peter Luschny, May 10 2016

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]    (* A054142 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A172431 *)

(* Clark Kimberling, Mar 09 2012 *)

Table[GegenbauerC[k-1, n-k+1, 1], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Dec 15 2019 *)

PROG

(PARI) T(n, k) = sum(j=0, (k-1)\2, (-1)^j*(n-j-1)!*2^(k-2*j-1)/(j!*(n-k)!*(k-2*j-1)!) );

for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 15 2019

(MAGMA) F:=Factorial; [ &+[(-1)^j*F(n-j-1)*2^(k-2*j-1)/(F(j)*F(n-k)*F(k-2*j-1)): j in [0..Floor((k-1)/2)]]: k in [1..n], n in [1..15]]; // G. C. Greubel, Dec 15 2019

(Sage) [[gegenbauer(k-1, n-k+1, 1) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Dec 15 2019

(GAP) F:=Factorial;; Flat(List([1..15], n-> List([1..n], k-> Sum([0..Int((k-1)/2)], j-> (-1)^j*F(n-j-1)*2^(k-2*j-1)/(F(j)*F(n-k)*F(k-2*j-1)) )))); # G. C. Greubel, Dec 15 2019

CROSSREFS

Cf. A078812, A053123, A007318, A001906 (antidiagonals sums), A007685.

Cf. also A054142, A082985.

Sequence in context: A093190 A132191 A094437 * A053123 A107661 A126570

Adjacent sequences:  A172428 A172429 A172430 * A172432 A172433 A172434

KEYWORD

nonn,tabl

AUTHOR

Mark Dols, Feb 02 2010

STATUS

approved

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Last modified November 30 20:17 EST 2021. Contains 349425 sequences. (Running on oeis4.)