A077023 -id:A077023 - cp's OEIS frontend

A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.

1, -1, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 0, 0, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 0, 0, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 0

Offset: 1

Clark Kimberling, Feb 03 2011

Using "->" to mean "is the weight array of" as defined at A144112:

A185779->A144225->A007318->A014430->A077023->A185779, where each of these is formatted as a rectangle (e.g., A007318 is Pascal's triangle). Read in reverse order, each is the accumulation array of the preceding array. It appears that successive weight arrays of A185779 contain Pascal's triangle except for initial terms.

			Northwest corner:
1....-1....0....0....0....0....0,...0
-1....2....0....0....0....0....0....0
0.....0....0....1....1....1....1....1
0.....0....1....2....3....4....5....6
0.....0....1....3....6....10...15...21
0.....0....1....4....10...20...35...56

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185779, A144225, A007318, A014430, A077023, A185779.

(* This code produces three arrays: A144225, A007318, A185778. *)
f[n_,0]:=0;f[0,k_]:=0;  (* Used to make the weight array *)
f[1,1]:=1;f[n_,1]:=0;f[1,k_]:=0
f[n_,2]:=1;f[2,k_]:=1;
f[n_,k_]:=-1+(n+k-4)!/((n-2)!*(k-2)!)/;k>1&&n>1;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A144225 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A007318, Pascal's triangle formatted as a rectangle *)
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185778 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

(See the Mathematica code.)

A185779 Third accumulation array of Pascal's triangle (as a rectangle), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 10, 17, 10, 20, 45, 45, 20, 35, 95, 126, 95, 35, 56, 175, 281, 281, 175, 56, 84, 294, 546, 662, 546, 294, 84, 120, 462, 966, 1358, 1358, 966, 462, 120, 165, 690, 1596, 2534, 2941, 2534, 1596, 690, 165, 220, 990, 2502, 4410, 5790, 5790, 4410, 2502, 990, 220, 286, 1375, 3762, 7272, 10620, 12021, 10620, 7272, 3762, 1375, 286, 364, 1859, 5467, 11484, 18432, 23229, 23229, 18432, 11484, 5467, 1859, 364, 455

Offset: 1

Clark Kimberling, Feb 03 2011

Using "Axxxxxx < Ayyyyyy" to mean that Ayyyyyy is the accumulation array of Axxxxxx, as defined at A144112:
A185779 < A144225 < A007318 < A014430 < A077023 < A185779, where each of these is formatted as a rectangle (e.g., A007318 is Pascal's triangle).  See A185778.
row 1: A000292
row 2: A095667

			Northwest corner:
1....4...10...20...35
4....17..45...95...175
10...45..126..281..546
20...95..281..662..1358

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A014430, A077023, A144112, A144225, A185778.

f[n_, k_] := Binomial[n + k + 4, n + 2] - (k + 3)*(k + 4)/2 - (k + 2)* n*(k*n + n + 3*k + 7)/4; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 5}]]
Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = C(n+k+4,n+2) - (k+3)*(k+4)/2 - (k+2)*n*(k*n+n+3*k+7)/4, for k>=1, n>=1.

A077022 Values of k such that sum of first k primes squared is divisible by k-th prime.

Original entry on oeis.org

1, 6, 17, 11156, 16548

Offset: 1

Randy L. Ekl and Zak Seidov, Oct 17 2002

Numbers k such that A072004(k) = 0.

a(6) > 10^11. - Giovanni Resta, Mar 07 2013

			6 is a term because A024450(6)/prime(6) = 29;
17 is a term because A024450(17)/prime(17) = 284;
11156 is a term because A024450(11156)/prime(11156) = 410066261;
16548 is a term because A024450(16548)/prime(16548) = 941945317.

Cf. A024450, A072004, A077023.

Module[{nn=17000,prs},prs=Accumulate[Prime[Range[nn]]^2];Select[Range[ nn],Divisible[prs[[#]],Prime[#]]&]] (* Harvey P. Dale, Nov 23 2013 *)

cp's OEIS Frontend

A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.

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A185779 Third accumulation array of Pascal's triangle (as a rectangle), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A077022 Values of k such that sum of first k primes squared is divisible by k-th prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica