A053445 -id:A053445 - cp's OEIS frontend

A147768 Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.

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

Offset: 0

Gary W. Adamson, Nov 11 2008

Row sums = A002865, (1, 0, 1, 1, 2, 2, 4, 4, 7, 8,...).

This triangle is the lower right half of a Toeplitz matrix. Each column of this triangle has the form [1, -1] U A053445. - Georg Fischer, Jul 28 2023

			First few rows of the triangle =
1;
-1, 1;
1, -1, 1;
0, 1, -1, 1;
1, 0, 1, -1, 1;
0, 1, 0, 1, -1, 1;
2, 0, 1, 0, 1, -1, 1;
0, 2, 0, 1, 0, 1, -1, 1;
3, 0, 2, 0, 1, 0, 1, -1, 1;
1, 3, 0, 2, 0, 1, 0, 1, -1, 1;
4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1;
2, 4, 1, 3, 0, 2, 0, 1, 0, 1, -1, 1;
...

Cf. A000012, A000041, A002865, A004736, A027293, A053445, A185018.

A000012^(-1) is the pairwise difference operator, and A027293 = a triangle with A000041 in every column.

Equals A185018 * A027293 since A000012^2 = A004736 and A004736^(-1) = A185018. - Georg Fischer, Jul 28 2023

A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 2, 1, 10, 14, 12, 6, 1, 15, 30, 39, 39, 20, 1, 21, 55, 95, 138, 142, 71, 1, 28, 91, 195, 364, 548, 551, 270, 1, 36, 140, 357, 804, 1564, 2317, 2278, 1100

Offset: 1

Alford Arnold, Apr 16 2009

The first five diagonals are essentially A000012, A000217, A000330, A086602 and A159571.
From Alford Arnold, Apr 20 2009: (Start)
After the first two diagonals, each additional diagonal is computed using blocks of source partitions (defined in A053445).
The size of each block increases by powers of two; e.g. 22, 33 222, 44 332 333 2222; etc.
Each source partition can be associated with a specific sequence as illustrated in the below example using partition 332: grow the leftmost value to form 432 then append "1" to form 3321. in like manner, generate 532 4321 and 33211 from the previously formed cases. Note that the number of arrangements are 3, 6+12, and 6+24+30 respectively and that we now have three terms of A006011: 3 18 and 60.
Next we note that 6 39 138 364 804 ... A159571 resulted from summing term by term, the sequences associated with partitions 44 332 333 and 2222:
1...5..14...30...55
3..18..60..150..315
1...7..25...65..140
1...9..39..119..294
(End)

Cf. A047970 (row sums), A000012, A000217, A000330, A086602, A159571.

Cf. A053445. - Alford Arnold, Apr 20 2009

Submitted without a definition. - N. J. A. Sloane, Apr 18 2009

More terms from Alford Arnold, Oct 06 2009

A182911 Prime encoded sequence of generic integer partitions of n in the antilexicographic order of the partitions.

Original entry on oeis.org

1, 2, 1, 1, 36, 1, 216, 900, 1, 1296, 5400, 44100, 27000, 7776, 32400, 264600, 5336100, 162000, 1323000, 46656, 194400, 810000, 1587600, 9261000, 32016600, 901800900, 972000, 7938000, 160083000, 279936, 1166400, 4860000, 9525600, 39690000, 55566000, 192099600, 1120581000

Offset: 0

Peter Luschny, Jan 26 2011

nonn

By definition [1] is a generic partition and 0 has no generic partitions. For n > 1 a partition p of n is generic if it does not have the form [1+r_1,r_2,...,r_k] or [r_1,r_2,...,r_k,1] for some partition [r_1,r_2,...,r_k] of n-1.
Encoding: The partition p = [p_1,...,p_k] is represented by Product_{i=1..k} prime(i) ^ p_i. If n has generic partitions then these encodings are listed in the antilexicographic order of the partitions; if n has no generic partitions then this fact is represented by '1'.
Starting from generic partitions a table of all partitions can be built by two operations: appending '1' at the tail of a partition or adding 1 to the head of a partition (see the table at the link given).
A generic partition is a partition of the form [x,x,p_2,...,p_k-1,y] with y > 1; in addition [1] is a generic partition by definition.

			0:  {}                   -> 1
1:  {[1]}                -> 2^1 = 2
2:  {}                   -> 1
3:  {}                   -> 1
4:  {[22]}               -> 2^2*3^2 = 36
5:  {}                   -> 1
6:  {[33],[222]}         -> 2^3*3^3 = 216; 2^2*3^2*5^2 = 900
7:  {}                   -> 1
8:  {[44],[332],[2222]}  -> 1296, 5400, 44100
9:  {[333]}              -> 27000

Alois P. Heinz, Table of n, a(n) for n = 0..19200
Peter Luschny, Integer partition trees, OEIS wiki.

Cf. A046056, A053445.

a:= proc(n) local b, ll; b:=
      proc(n,i,l) local nl; nl:= nops(l);
        if n<0 then
      elif n=0 then ll:= ll,
               `if`(nl=0 or nl=1 and l[1]=1 or
                    nl>1 and l[-1]<>1 and l[1]=l[2],
                    mul(ithprime(t)^l[t], t=1..nl), NULL)
      elif i=0 then
      else b(n-i, i, [l[], i]), b(n, i-1, l)
        fi
      end;
      ll:= NULL; b(n,n,[]);
     `if`(ll=NULL,1,ll)
    end:
seq(a(n), n=0..15);

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280

Offset: 0

Gus Wiseman, Jan 12 2025

sign

Also the inverse zero-based binomial transform of the odd prime numbers.

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

A162934 Shift sequence A162932 twice then subtract from the original sequence.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 4, 2, 2, 5, 3, 4, 9, 5, 6, 13, 11, 10, 19, 17, 19, 28, 27, 31, 44, 41, 49, 66, 68, 74, 98, 104, 118, 145, 157, 178, 220, 234, 268, 322, 354, 397, 473, 521, 591, 686, 765, 863, 1003, 1107, 1254, 1444, 1609

Offset: 6

Alford Arnold, Aug 05 2009, Aug 06 2009

From Alford Arnold, Dec 17 2009: (Start)
At n = 24, six of the partitions can be associated with the sixth row of this triangular array:
 333
 444 3333
 555 4443 33333
 666 5553 44433 333333
 777 6663 55533 444333 3333333
 888 7773 66633 555333 4443333 33333333
The other three partitions are new; and hence on their first row, so 6*1 + 1*3 = 9.
In a similar manner, the 44 cases at n = 36 can be computed using the array row numbers and the number of applicable partitions. Thus we have:
(10, 5, 3, 2, 1) times (1, 3, 2, 3, 7) providing 10 + 15 + 6 + 6 + 7 = 44 cases. (End)

			For n = 24, the sequence counts these nine partitions of 24: 888, 7773, 66633, 55554, 555333, 4443333, 6666, 444444, 33333333.

Cf. A000041, A002865, A053445, A162932.

G.f.: Sum_{n >= 0} q^(3*n+6)/Product_{k = 1..n} 1 - q^(k+2). - Peter Bala, Dec 01 2024

More terms from Alford Arnold, Dec 17 2009

More terms from Joerg Arndt, Jul 16 2015

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A147768 Triangle read by rows: A000012^(-2) * A027293 as infinite lower triangular matrices.

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A159572 A triangular array related to ordered partitions and having row sums 1,2,5,14,43,144,523,2048,8597... A047970.

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A182911 Prime encoded sequence of generic integer partitions of n in the antilexicographic order of the partitions.

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A379542 Second term of the n-th differences of the prime numbers.

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A162934 Shift sequence A162932 twice then subtract from the original sequence.

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