A211233
Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 3, n >= 1.
Original entry on oeis.org
1, 2, 3, 1, 4, 10, 4, 1, 1, 7, 27, 13, -13, -27, -7, -1, 1, 12, 69, 16, -182, -376, -182, 16, 69, 12, 1, 1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, -176, -21, -1, 1, 38, 456, -886, -8292, -20322, -6536, 35890, 65862, 35890, -6536, -20322, -8292, -886, 456, 38, 1
Offset: 1
Triangle begins
1, 2, 3;
1, 4, 10, 4, 1;
1, 7, 27, 13, -13, -27, -7, -1;
1, 12, 69, 16, -182, -376, -182, 16, 69, 12, 1;
1, 21, 176, -88, -1375, -3123, -1608, 1608, 3123, 1375, 88, ... ;
...
Row sums of even rows are
A047682; row sums of odd rows are zero for n > 1.
-
T(n,r=3)={my(R=vector(n)); R[1]=[1..r]; for(n=2, n, my(u=R[n-1]); R[n]=vector(r*n-1, k, sum(j=0, r, (k - j*n)*if(k>j && k-j<=#u, u[k-j], 0)))); R}
{my(A=T(5)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, May 18 2020
A211234
Triangle read by rows: T(n,k) is the k-th generalized Eulerian number of order n and degree 4, n >= 1.
Original entry on oeis.org
1, 2, 3, 4, 1, 4, 10, 20, 10, 4, 1, 1, 7, 27, 77, 57, 0, -57, -77, -27, -7, -1, 1, 12, 69, 272, 221, -272, -1084, -1688, -1084, -272, 221, 272, 69, 12, 1, 1, 21, 176, 936, 625, -3288, -11868, -21023, -16223, 0, 16223, 21023, 11868, 3288, -625, -936, -176, -21, -1
Offset: 1
Triangle begins:
1, 2, 3, 4;
1, 4, 10, 20, 10, 4, 1;
1, 7, 27, 77, 57, 0, -57, -77, -27, -7, -1;
...
Row sums of even rows are
A047683; row sums of odd rows are zero for n > 1.
-
T(n,r=4)={my(R=vector(n)); R[1]=[1..r]; for(n=2, n, my(u=R[n-1]); R[n]=vector(r*n-1, k, sum(j=0, r, (k - j*n)*if(k>j && k-j<=#u, u[k-j], 0)))); R}
{ my(A=T(5)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, May 18 2020
A211235
Array of generalized Eulerian numbers C(n,k) read by antidiagonals.
Original entry on oeis.org
1, 1, 2, 1, 4, 3, 1, 7, 10, 4, 1, 12, 27, 20, 5, 1, 21, 69, 77, 35, 6, 1, 38, 176, 272, 182, 56, 7, 1, 71, 456, 936, 846, 378, 84, 8, 1, 136, 1205, 3210, 3750, 2232, 714, 120, 9, 1, 265, 3247, 11075, 16290, 12342, 5214, 1254, 165, 10
Offset: 1
Array begins:
1, 2, 3, 4, 5, 6, ... A000027
1, 4, 10, 20, 35, 56, ... A000292
1, 7, 27, 77, 182, 378, ... A005585
1, 12, 69, 272, 846, 2232, ... A101097
1, 21, 176, 936, 3750, 12342, ... A254681
A005126,
...
Triangle begins:
1
1 2
1 4 3
1 7 10 4
1 12 27 20 5
1 21 69 77 35 6
1 38 176 272 182 56 7
...
-
A211235 := (n, k) -> add(binomial(n-i, k-i)*i^(n-k), i = 1 .. k): for n from 1 to 10 do seq(A211235(n, k), k = 1 .. n) end do; # Peter Bala, Oct 27 2015
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T[n_, k_] := Sum[Binomial[n-i, k-i] * i^(n-k), {i, 1, k}]; Table[T[n, k], {n,1,10}, {k,1,n}] //Flatten (* Amiram Eldar, Nov 30 2018 *)
Terms a(37)-a(55) added by
Peter Bala, Oct 27 2015
Showing 1-3 of 3 results.