A070936 -id:A070936 - cp's OEIS frontend

A070937 Number of times maximal coefficient (A025591) appears in Product_{k<=n} (x^k + 1), i.e., number of times highest value appears in n-th row of A053632 or n-th column of A070936.

1, 2, 4, 1, 5, 6, 4, 5, 1, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1

Offset: 0

Henry Bottomley, May 12 2002

			a(4)=5 since Product_{k<=4} (x^k + 1) = 1 + x + x^2 + 2x^3 + 2x^4 + 2x^5 + 2x^6 + 2x^7 + x^8 + x^9 + x^10 and 2 appears as a coefficient 5 times.

Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

If n mod 4 = 0 or 3 then a(n) odd, otherwise a(n) even.
For n > 9: a(n) = A014695(n).
From Chai Wah Wu, Apr 10 2021: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 12.
G.f.: (2*x^12 - 2*x^11 + 6*x^10 - 4*x^9 + 6*x^8 - 2*x^7 - 2*x^6 + 2*x^5 - 6*x^4 + 2*x^3 - 3*x^2 - x - 1)/((x - 1)*(x^2 + 1)). (End)

A026836 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the greatest being k, for k=1,2,...,n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1

Offset: 1

Clark Kimberling

Conjecture: A199918(n) = Sum_{k=1..n} (-1)^(n-k) T(n,k). - George Beck, Jan 13 2019

			Triangle begins:
[1]
[0, 1]
[0, 1, 1]
[0, 0, 1, 1]
[0, 0, 1, 1, 1]
[0, 0, 1, 1, 1, 1]
[0, 0, 0, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 1, 3, 4, 4, 3, 2, 2, 1, 1, 1]
... - _N. J. A. Sloane_, Nov 09 2018

Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

If seen as a square array then transpose of A070936 and visible form of A053632. Central diagonal and those to the right of center are A000009 as are row sums.

with(combinat);
f2:=proc(n) local i,j,p,t0,t1,t2;
t0:=Array(1..n,0);
t1:=partition(n);
p:=numbpart(n);
for i from 1 to p do
t2:=t1[i];
if nops(convert(t2,set))=nops(t2) then
# now have a partition t2 of n into distinct parts
t0[t2[-1]]:=t0[t2[-1]]+1;
od:
[seq(t0[j],j=1..n)];
end proc;
for n from 1 to 12 do lprint(f2(n)); od: # N. J. A. Sloane, Nov 09 2018

T(n, k) = A070936(n-k, k-1) = A053632(k-1, n-k) = T(n-1, k-1)+T(n-2k+1, k-1). - Henry Bottomley, May 12 2002

T(n, k) = coefficient of x^n in x^k*Product_{i=1..k-1} (1+x^i). - Vladeta Jovovic, Aug 07 2003

A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 13, 1, 0, 1, 1, 3, 19, 49, 1, 0, 1, 1, 3, 19, 97, 261, 1, 0, 1, 1, 3, 19, 121, 681, 1531, 1, 0, 1, 1, 3, 19, 121, 921, 5971, 9073, 1, 0, 1, 1, 3, 19, 121, 1041, 8491, 50443, 63393, 1, 0, 1, 1, 3, 19, 121, 1041

Offset: 0

Seiichi Manyama, Oct 26 2017

			Square array A(n,k) begins:
   1, 1,   1,   1,   1, ...
   0, 1,   1,   1,   1, ...
   0, 1,   3,   3,   3, ...
   0, 1,  13,  19,  19, ...
   0, 1,  49,  97, 121, ...
   0, 1, 261, 681, 921, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000012, A118589, A294251, A294252, A294253.
Rows n=0 gives A000012.
Main diagonal gives A293840.
Cf. A070936, A294212, A294254.

B(j,k) is the coefficient of Product_{i=1..k} (1+x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(A000217(k),n)} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

cp's OEIS Frontend

A070937 Number of times maximal coefficient (A025591) appears in Product_{k<=n} (x^k + 1), i.e., number of times highest value appears in n-th row of A053632 or n-th column of A070936.

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A026836 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the greatest being k, for k=1,2,...,n.

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A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

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