A118981 -id:A118981 - cp's OEIS frontend

A309220 Square array A read by antidiagonals: the columns are given by A(n,1)=1, A(n,2)=n+1, A(n,3) = n^2+2n+3, A(n,4) = n^3+3n^2+6n+4, A(n,5) = n^4+4n^3+10n^2+12*n+7, ..., whose coefficients are given by A104509 (see also A118981).

1, 1, 2, 1, 3, 6, 1, 4, 11, 14, 1, 5, 18, 36, 34, 1, 6, 27, 76, 119, 82, 1, 7, 38, 140, 322, 393, 198, 1, 8, 51, 234, 727, 1364, 1298, 478, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 1, 11, 102, 756, 4354, 18557

Offset: 1

N. J. A. Sloane, Aug 12 2019, based on R. J. Mathar's 2011 analysis of A118980

As pointed out by Peter Munn, A117938 gives the same triangle, except that it has an additional diagonal at the right. - N. J. A. Sloane, Aug 13 2019

			The first few antidiagonals are:
1,
1,2,
1,3,6,
1,4,11,14,
1,5,18,36,34,
1,6,27,76,119,82,
1,7,38,140,322,393,198,
...
The first nine rows of A are
1, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...
1, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, 510117, ...
1, 4, 18, 76, 322, 1364, 5778, 24476, 103682, 439204, 1860498, 7881196, ...
1, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, 73997555, ...
1, 6, 38, 234, 1442, 8886, 54758, 337434, 2079362, 12813606, 78960998, 486579594, ...
1, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, 48229636, 344362251, 2458765393, ...
1, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, 153992264, 1250895426, 10161155672, ...
1, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, 432083484, 3936182123, 35857722591, ...
1, 10, 102, 1030, 10402, 105050, 1060902, 10714070, 108201602, 1092730090, 11035502502, 111447755110, ...

Cf. A104509, A117938, A118980, A118981, A099425 (top row), A006497 (essentially the 2nd row), A014448 (essentially the 3rd row), A087130 (essentially the 4th row).

M := 12;
A:=Array(1..2*M,1..2*M,0):
for i from 1 to M do A[i,1]:=1; od:
S := series((1 + x^2)/(1-x-x^2 + x*y), x, 120): # this is g.f. for A104509
for n from 1 to M do
R2 := expand(coeff(S, x, n));
R3 := [seq(abs(coeff(R2,y,n-i)),i=0..n)];
f := k-> add( R3[i]*k^(n-i+1), i=1..nops(R3) ): # this is the formula for the (n+1)-st column
s1 := [seq(f(i),i=1..M)];
for i from 1 to M do A[i,n+1]:=s1[i]; od:
od:
for i from 1 to M do lprint([seq(A[i,j],j=1..M)]); od:
# alternative by R. J. Mathar, Aug 13 2019 :
A104509 := proc(n,k)
    (1+x^2)/(1-x-x^2+x*y) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,k) ;
end proc:
A309220 := proc(n::integer,k::integer)
    local x;
    add( abs(A104509(k-1,i))*x^i,i=0..k-1) ;
    subs(x=n,%) ;
end proc:
seq( seq(A309220(d-k,k),k=1..d-1),d=2..13) ;

A117938 Triangle, columns generated from Lucas Polynomials.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 11, 14, 7, 1, 5, 18, 36, 34, 11, 1, 6, 27, 76, 119, 82, 18, 1, 7, 38, 140, 322, 393, 198, 29, 1, 8, 51, 234, 727, 1364, 1298, 478, 47, 1, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 1, 10, 83, 536, 2599, 8886, 19602, 24476, 14159, 2786, 123

Offset: 1

Gary W. Adamson, Apr 03 2006

Companion triangle using Fibonacci polynomial generators = A073133. Inverse binomial transforms of the columns defines rows of A117937 (with some adjustments of offset).

A309220 is another version of the same triangle (except it omits the last diagonal), and perhaps has a clearer definition. - N. J. A. Sloane, Aug 13 2019

			First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  6,   4;
  1, 4, 11,  14,   7;
  1, 5, 18,  36,  34,  11;
  1, 6, 27,  76, 119,  82,  18;
  1, 7, 38, 140, 322, 393, 198, 29;
  ...
For example, T(7,4) = 76 = f(4), x^3 + 3*x = 64 + 12 = 76.

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

Cf. A104509, A114525, A117936, A117937, A118980, A118981, A309220.

Cf. A000204 (diagonal), A059100 (column 3), A061989 (column 4).

Lucas := proc(n,x) # see A114525
    option remember;
    if  n=0 then
        2;
    elif n =1 then
        x ;
    else
        x*procname(n-1,x)+procname(n-2,x) ;
    end if;
    expand(%) ;
end proc:
A117938 := proc(n::integer,k::integer)
    if k = 1 then
        1;
    else
        subs(x=n-k+1,Lucas(k-1,x)) ;
    end if;
end proc:
seq(seq(A117938(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 16 2019

T[n_, k_]:= LucasL[k-1, n-k+1] - Boole[k==1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 28 2021 *)

def A117938(n,k): return 1 if (k==1) else round(2^(1-k)*( (n-k+1 + sqrt((n-k)*(n-k+2) + 5))^(k-1) + (n-k+1 - sqrt((n-k)*(n-k+2) + 5))^(k-1) ))
flatten([[A117938(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 28 2021

Columns are f(x), x = 1, 2, 3, ..., of the Lucas Polynomials: (1, defined different from A034807 and A114525); (x); (x^2 + 2); (x^3 + 3*x); (x^4 + 4*x^2 + 2); (x^5 + 5*x^3 + 5*x); (x^6 + 6*x^4 + 9*x^2 + 2); (x^7 + 7*x^5 + 14*x^3 + 7*x); ...

Terms a(51) and a(52) corrected by G. C. Greubel, Oct 28 2021

A118980 Triangle read by rows: rows = inverse binomial transforms of columns of A309220.

Original entry on oeis.org

1, 2, 1, 6, 5, 2, 14, 22, 18, 6, 34, 85, 118, 84, 24, 82, 311, 660, 780, 480, 120, 198, 1100, 3380, 5964, 6024, 3240, 720, 478, 3809, 16380, 40740, 60480, 52920, 25200, 5040, 1154, 13005, 76518, 258804, 531864, 676080, 519840, 221760, 40320, 2786, 43978, 348462, 1564314, 4286880, 7444800, 8240400

Offset: 1

Gary W. Adamson, May 07 2006

First few columns of A309220:
  1, 2,  6,  14,   34, ...
  1, 3, 11,  36,  119, ...
  1, 4, 18,  76,  322, ...
  1, 5, 27, 140,  727, ...
  1, 6, 38, 234, 1442, ...
  1, 7, 51, 364, 2599, ...
  1, 8, 66, 536, 4354, ...
  ...

			First few rows of the triangle:
   1;
   2,   1;
   6,   5,   2;
  14,  22,  18,   6;
  34,  85, 118,  84,  24;
  82, 311, 660, 780, 480, 120;
  ...
Column 3 of A309220 = (6, 11, 18, 27, 38, 51, ...), whose inverse binomial transform is (6, 5, 2).

The leading column is A099425, and the rightmost two diagonals are A038720 and A000142.

Cf. A104509, A117938, A118981, A309220.

with(transforms);
M := 12;
T := [1];
S := series((1 + x^2)/(1-x-x^2 + x*y), x, 120):
for n from 1 to M do
R2 := expand(coeff(S, x, n));
R3 := [seq(abs(coeff(R2,y,n-i)),i=0..n)];
f := k-> add( R3[i]*k^(n-i+1), i=1..nops(R3) ):
s1 := [seq(f(i),i=1..3*n)];
s2 := BINOMIALi(s1);
s3 := [seq(s2[i],i=1..n+1)];
T := [op(T), op(s3)];
od:
T;  # N. J. A. Sloane, Aug 12 2019

Edited and extended by N. J. A. Sloane, Aug 12 2019, guided by the comments of R. J. Mathar from Oct 30 2011

cp's OEIS Frontend

A309220 Square array A read by antidiagonals: the columns are given by A(n,1)=1, A(n,2)=n+1, A(n,3) = n^2+2n+3, A(n,4) = n^3+3n^2+6n+4, A(n,5) = n^4+4n^3+10n^2+12*n+7, ..., whose coefficients are given by A104509 (see also A118981).

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A117938 Triangle, columns generated from Lucas Polynomials.

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A118980 Triangle read by rows: rows = inverse binomial transforms of columns of A309220.

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cp's OEIS Frontend

A309220 Square array A read by antidiagonals: the columns are given by A(n,1)=1, A(n,2)=n+1, A(n,3) = n^2+2n+3, A(n,4) = n^3+3*n^2+6*n+4, A(n,5) = n^4+4*n^3+10*n^2+12*n+7, ..., whose coefficients are given by A104509 (see also A118981).

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Author

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Examples

Crossrefs

Programs

Maple

A117938 Triangle, columns generated from Lucas Polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A118980 Triangle read by rows: rows = inverse binomial transforms of columns of A309220.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A309220 Square array A read by antidiagonals: the columns are given by A(n,1)=1, A(n,2)=n+1, A(n,3) = n^2+2n+3, A(n,4) = n^3+3n^2+6n+4, A(n,5) = n^4+4n^3+10n^2+12*n+7, ..., whose coefficients are given by A104509 (see also A118981).