A212362 - OEIS

%I #31 Sep 08 2022 08:46:02

%S 1,2,2,4,7,3,8,19,15,4,16,47,52,26,5,32,111,155,110,40,6,64,255,426,

%T 385,200,57,7,128,575,1113,1211,805,329,77,8,256,1279,2808,3556,2856,

%U 1498,504,100,9,512,2815,6903,9948,9324,5922,2562,732,126,10

%N Triangle by rows, binomial transform of the beheaded Pascal's triangle A074909.

%C Row sums of the triangle inverse = A027641/A027642, the Bernoulli numbers; (1, -1/2, 1/6, 0, -1/30,...)

%H G. C. Greubel, <a href="/A212362/b212362.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Binomial transform of the beheaded Pascal's triangle (A074909) as a matrix. (The beheaded Pascal matrix deletes the rightmost border of 1's.)

%F From _G. C. Greubel_, Aug 05 2021: (Start)

%F T(n, k) = Sum_{j=0..n} binomial(n, j)*binomial(j+1, k) - binomial(n, k-1), with T(n, 0) = 2^n.

%F T(n, k) = 2^(n-k)*binomial(n+1, k) + (2^(n-k) - 1)*binomial(n, k-1).

%F Sum_{k=0..n} T(n, k) = A027649(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A106515(n). (End)

%e First few rows of the triangle are:

%e     1;

%e     2,    2;

%e     4,    7,    3;

%e     8,   19,   15,    4

%e    16,   47,   52,   26,    5;

%e    32,  111,  155,  110,   40,    6;

%e    64,  255,  426,  385,  200,   57,   7;

%e   128,  575, 1113, 1211,  805,  329,  77,   8;

%e   256, 1279, 2808, 3556, 2856, 1498, 504, 100, 9;

%e   ...

%p A212362 := proc(n,k)

%p         add( binomial(n,i)*A074909(i,k),i=0..n) ;

%p end proc: # _R. J. Mathar_, Aug 03 2015

%t T[n_, k_]= 2^(n-k)*Binomial[n+1, k] + (2^(n-k) -1)*Binomial[n, k-1];

%t Table[T[n, k] , {n,0,12}, {k,0,n}] //Flatten (* _G. C. Greubel_, Aug 05 2021 *)

%o (Magma)

%o A074909:= func< n,k | k lt 0 or k gt n select 0 else Binomial(n+1, k) >;

%o A212362:= func< n,k | (&+[ Binomial(n,j)*A074909(j, k) : j in [0..n]]) >;

%o [A212362(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 05 2021

%o (Sage)

%o def T(n, k): return 2^(n-k)*binomial(n+1, k) + (2^(n-k) - 1)*binomial(n, k-1)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Aug 05 2021

%Y Cf. A074909, A027641/A027642, A027649 (row sums), A006589 (2nd column), A106515.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Jun 29 2012

%E a(22) corrected by _G. C. Greubel_, Aug 05 2021