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A135230 Triangle A103451 * A000012(signed) * A007318, read by rows. 2
1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 2, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 2, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1, 2, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

row sums = A135231

LINKS

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

FORMULA

T(n,k) = A103451 * A000012(signed) * A007318, where A000012(signed) = (1; -1,1; 1,-1,1;...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = (3+(-1)^n)/2 and T(n,n) = 1. - G. C. Greubel, Nov 20 2019

EXAMPLE

First few rows of the triangle are:

1;

1, 1;

2, 1, 1;

1, 2, 2, 1;

2, 2, 4, 3, 1;

1, 3, 6, 7, 4, 1;

2, 3, 9, 13, 11, 5, 1;

1, 4, 12, 22, 24, 16, 6, 1;

2, 4, 16, 34, 46, 40, 22, 7, 1;

...

MAPLE

T:= proc(n, k) option remember;

if k=n then 1

elif k=0 then (3+(-1)^n)/2

else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))

fi; end:

seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

MATHEMATICA

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3+(-1)^n)/2, Sum[Binomial[n-1 - 2*j, k-1], {j, 0, Floor[(n-1)/2]}] ]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

PROG

(PARI) T(n, k) = if(k==n, 1, if(k==0, (3+(-1)^n)/2, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) )); \\ G. C. Greubel, Nov 20 2019

(Magma)

function T(n, k)

if k eq n then return 1;

elif k eq 0 then return (3+(-1)^n)/2;

else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);

end if; return T; end function;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

(Sage)

@CachedFunction

def T(n, k):

if (k==n): return 1

elif (k==0): return (3+(-1)^n)/2

else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))

[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

CROSSREFS

Cf. A000012, A007318, A103451, A135231.

Sequence in context: A086995 A220492 A229873 * A117957 A145702 A145704

Adjacent sequences: A135227 A135228 A135229 * A135231 A135232 A135233

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Nov 23 2007

EXTENSIONS

More terms and offset changed by G. C. Greubel, Nov 20 2019

STATUS

approved

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Last modified January 28 20:13 EST 2023. Contains 359905 sequences. (Running on oeis4.)