A000712 -id:A000712 - cp's OEIS frontend

A348617 Numbers whose sum of prime indices is twice their negated alternating sum.

1, 10, 39, 88, 115, 228, 259, 306, 517, 544, 620, 783, 793, 870, 1150, 1204, 1241, 1392, 1656, 1691, 1722, 1845, 2369, 2590, 2596, 2775, 2944, 3038, 3277, 3280, 3339, 3498, 3692, 3996, 4247, 4440, 4935, 5022, 5170, 5226, 5587, 5644, 5875, 5936, 6200, 6321

Offset: 1

Gus Wiseman, Nov 26 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their negated alternating sum.

			The terms and their prime indices begin:
     1: ()
    10: (3,1)
    39: (6,2)
    88: (5,1,1,1)
   115: (9,3)
   228: (8,2,1,1)
   259: (12,4)
   306: (7,2,2,1)
   517: (15,5)
   544: (7,1,1,1,1,1)
   620: (11,3,1,1)
   783: (10,2,2,2)
   793: (18,6)
   870: (10,3,2,1)
  1150: (9,3,3,1)
  1204: (14,4,1,1)
  1241: (21,7)
  1392: (10,2,1,1,1,1)
  1656: (9,2,2,1,1,1)
  1691: (24,8)

These partitions are counted by A001523 up to 0's.
An ordered version is A349154, nonnegative A348614, reverse A349155.
The nonnegative version is A349159, counted by A000712 up to 0's.
The reverse nonnegative version is A349160, counted by A006330 up to 0's.
A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
A034871, A097805, A345197 count compositions by alternating sum.
A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A103919 counts partitions by alternating sum, reverse A344612.
A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000984, A001700, A028260, A045931, A120452, A195017, A257991, A257992, A262977, A325698, A344619, A345958.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[1000],Total[primeMS[#]]==-2*ats[primeMS[#]]&]

A056239(a(n)) = -2*A316524(a(n)).

A346698(a(n)) = 3*A346697(a(n)).

A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

Original entry on oeis.org

0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276

Offset: 1

Gus Wiseman, Nov 21 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms and corresponding compositions begin:
       0: ()
      12: (1,3)
     160: (2,6)
     193: (1,6,1)
     195: (1,5,1,1)
     198: (1,4,1,2)
     204: (1,3,1,3)
     216: (1,2,1,4)
     240: (1,1,1,5)
    2304: (3,9)
    2561: (2,9,1)
    2563: (2,8,1,1)
    2566: (2,7,1,2)
    2572: (2,6,1,3)
    2584: (2,5,1,4)

These compositions are counted by A224274 up to 0's.
Except for 0, a subset of A345919.
The positive version is A348614, reverse A349153.
An unordered version is A348617, counted by A001523.
The reverse version is A349155.
A positive unordered version is A349159, counted by A000712 up to 0's.
A000346 = even-length compositions with alt sum != 0, complement A001700.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Necklaces are ranked by A065609, dual A333764, reversed A333943.
- Alternating compositions are ranked by A345167, complement A345168.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==-2*ats[stc[#]]&]

A052837 Number of partitions of 2n whose Ferrers-Young diagram allows more than one different domino tiling.

Original entry on oeis.org

0, 0, 1, 4, 10, 22, 43, 80, 141, 240, 397, 640, 1011, 1568, 2395, 3604, 5360, 7876, 11460, 16510, 23588, 33418, 47006, 65640, 91085, 125596, 172215, 234820, 318579, 430060, 577920, 773130, 1030007, 1366644, 1806445, 2378892, 3121835, 4082796, 5322360, 6916360

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The original name was: A simple grammar.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 804

Essentially the same as A048574.

Cf. A000712, A304789.

spec := [S,{C=Sequence(Z,1 <= card),B=Set(C,1 <= card),S=Prod(B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
a:= n-> (p-> add(p(j)*p(n-j), j=1..n-1))(combinat[numbpart]):
seq(a(n), n=0..40);  # Alois P. Heinz, May 26 2018

a[n_] := If[n <= 1, 0, With[{pp = Array[PartitionsP, n-1]},
   First[ListConvolve[pp, pp]]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 30 2025 *)

G.f.: (exp(Sum_{j>=1} -x^j/((x^j-1)*j) )-1)^2.

a(n) = Sum_{k>=2} A304789(n,k). - Alois P. Heinz, May 26 2018

More terms from Franklin T. Adams-Watters, Feb 08 2006

New name from Alois P. Heinz, May 26 2018

A061161 Numerators in expansion of Euler transform of b(n) = 1/4.

Original entry on oeis.org

1, 1, 13, 55, 1235, 4615, 55801, 200343, 8977475, 36804235, 367235363, 1444888289, 32062742231, 120729974115, 1205864254225, 5201022002071, 395884671433315, 1603069490974835, 15989295873680415, 64312573140322525, 1250332447587844829, 5262481040435242585

Offset: 0

Vladeta Jovovic, Apr 17 2001

Denominators of c(n) are 2^A004134(n).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Geoffrey B. Campbell and A. Zujev, Some almost partition theoretic identities, Preprint, 2016.
N. J. A. Sloane, Transforms

Cf. A000712, A061159, A061160.

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d/4, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 28 2017

c[n_] := c[n] = If[n == 0, 1,
     (1/(4n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
a[n_] := Numerator[c[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2022 *)

Numerators of c(n), where c(n) = (1/(4*n))*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.

A103929 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 10.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 751, 1162, 1762, 2647, 3918, 5748, 8331, 11981, 17056, 24108, 33787, 47043, 65019, 89336, 121954, 165585, 223542, 300295, 401331, 533937, 707057, 932404, 1224376, 1601571, 2086851, 2709449, 3505228

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).

In general, column m of A008951 is asymptotic to exp(Pi*sqrt(2*n/3)) * 6^(m/2) * n^((m-2)/2) / (4*sqrt(3) * m! * Pi^m), equivalently to 6^(m/2) * n^(m/2) / (m! * Pi^m) * p(n), where p(n) is the partition function A000041. - Vaclav Kotesovec, Aug 28 2015

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 91.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Eleventh column (m=10) of Fine-Riordan triangle A008951 and of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.

Cf. A000712 (all parts of two kinds).

nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 10}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@10], {n,0,37}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 55, 10];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (product(1/(1-x^k), k=1..10)^2)*product(1/(1-x^j), j=11..infty).
a(n)=sum(A103924(n-10*j), j=0..floor(n/10)), n>=0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^5 * n^4 / (4*sqrt(3) * 10! * Pi^10). - Vaclav Kotesovec, Aug 28 2015

A108803 A108802 read mod 4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 09 2005

nonn

A000041 := proc(n) combinat[numbpart](n) ; end: A040051 := proc(n) option remember ; A000041(n) mod 2 ; end: A108802 := proc(n) option remember ; add( A040051(i)*A040051(n-i-1),i=0..n-1) ; end: A108803 := proc(n) A108802(n) mod 4 ; end: for n from 1 to 140 do printf("%d, ",A108803(n)) ; od ; # R. J. Mathar, May 08 2007

Function[w, Mod[ListConvolve[#, #], 4] & /@ Map[Take[w, #] &, Range@ Length@ w]]@ Table[Mod[PartitionsP@ n, 2], {n, 0, 105}] // Flatten (* Michael De Vlieger, Jul 16 2016 *)

a(n) = A000712(n-1)(mod 4). - John M. Campbell, Jul 16 2016

More terms from R. J. Mathar, May 08 2007

A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 19, 38, 35, 19, 6, 1, 1, 30, 74, 86, 59, 26, 7, 1, 1, 45, 139, 194, 164, 91, 34, 8, 1, 1, 67, 249, 415, 416, 281, 132, 43, 9, 1, 1, 97, 434, 844, 990, 787, 447, 183, 53, 10, 1

Offset: 0

Omar E. Pol, Jun 27 2012

It appears that row 2 is A034856.
Observation:
Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...
Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...

			Array begins:
1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
1,   4,   8,  13,  19,  26,  34,  43,  53,
1,   7,  18,  35,  59,  91, 132, 183,
1,  12,  38,  86, 164, 281, 447,
1,  19,  74, 194, 416, 787,
1,  30, 139, 415, 990,
1,  45, 249, 844,
1,  67, 434,
1,  97,
1,

Alois P. Heinz, Rows n = 0..140, flattened

Columns (0-3): A000012, A000070, A000713, A210843.
Rows (0-1): A000012, A000027.
Main diagonal gives A303070.
Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.

with(numtheory):
etr:= proc(p) local b;
        b:= proc(n) option remember; `if`(n=0, 1,
              add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)
            end
      end:
A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):
seq(seq(A(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, May 20 2013

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[{j}, k + If[j == 1, 1, 0]]][n]; Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A292577 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^jjx^(j*i))^2.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 5, 0, 1, 2, 1, 10, 0, 1, 2, 1, -2, 20, 0, 1, 2, 1, 4, -4, 36, 0, 1, 2, 1, 4, 14, 4, 65, 0, 1, 2, 1, 4, 6, 16, 13, 110, 0, 1, 2, 1, 4, 6, -8, 10, 6, 185, 0, 1, 2, 1, 4, 6, 2, -6, 42, -23, 300, 0, 1, 2, 1, 4, 6, 2, 24, 18, 109, -44, 481, 0, 1

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0,  2,  2,  2,  2, ...
   0,  5,  1,  1,  1, ...
   0, 10, -2,  4,  4, ...
   0, 20,  4, 14,  6, ...
   0, 36, 13, 16, -8, ...

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

Columns k=0..1 give A000007, A000712.
Rows n=0 gives A000012.
Main diagonal gives A293387.
Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^j*j*x^(j*i))^m: this sequence (m=-2), A293307 (m=-1), A293305 (m=1), A293388 (m=2).

A298436 Expansion of Product_{k>=1} 1/(1 - x^prime(k))^2.

Original entry on oeis.org

1, 0, 2, 2, 3, 6, 7, 12, 15, 22, 30, 40, 54, 72, 93, 122, 157, 202, 256, 326, 409, 512, 640, 792, 981, 1204, 1479, 1802, 2196, 2662, 3218, 3880, 4660, 5588, 6677, 7960, 9471, 11232, 13299, 15710, 18514, 21784, 25570, 29968, 35047, 40922, 47698, 55500, 64480, 74786, 86618

Offset: 0

Ilya Gutkovskiy, Jan 19 2018

nonn

Number of partitions of n into prime parts of 2 kinds.

Self-convolution of A000607.

			a(5) = 6 because we have [5a], [5b], [3a, 2a], [3a, 2b], [3b, 2a] and [3b, 2b].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Index entries for related partition-counting sequences

Cf. A000040, A000607, A000712, A280245.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^prime(k))^2.

log(a(n)) ~ 2*Pi*sqrt(2*n/(3*log(n/2))). - Vaclav Kotesovec, Jan 12 2021

A304796 Number of special sums of integer partitions of n.

Original entry on oeis.org

1, 2, 5, 10, 18, 32, 51, 82, 122, 188, 262, 392, 529, 750, 997, 1404, 1784, 2452, 3123, 4164, 5239, 6916, 8499, 11112, 13693, 17482, 21257, 27162, 32581, 41114, 49606, 61418, 73474, 91086, 107780, 132874, 157359, 191026, 225159, 274110, 320691, 386722, 453875

Offset: 0

Gus Wiseman, May 18 2018

nonn

A special sum of an integer partition y is a number n >= 0 such that exactly one submultiset of y sums to n.

			The a(4) = 18 special positive subset-sums:
0<=(4), 4<=(4),
0<=(22), 2<=(22), 4<=(22),
0<=(31), 1<=(31), 3<=(31), 4<=(31),
0<=(211), 1<=(211), 3<=(211), 4<=(211),
0<=(1111), 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111).

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854.

uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

a(n) = A301854(n) + A000041(n).

More terms from Alois P. Heinz, May 18 2018

a(36)-a(42) from Chai Wah Wu, Sep 26 2023

cp's OEIS Frontend

A348617 Numbers whose sum of prime indices is twice their negated alternating sum.

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A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

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A052837 Number of partitions of 2n whose Ferrers-Young diagram allows more than one different domino tiling.

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A061161 Numerators in expansion of Euler transform of b(n) = 1/4.

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A103929 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 10.

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A108803 A108802 read mod 4.

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A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

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A292577 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^j*j*x^(j*i))^2.

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A292577 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^jjx^(j*i))^2.