A262977 -id:A262977 - cp's OEIS frontend

A357487 Number of integer partitions of n with the same length as reverse-alternating sum.

1, 1, 0, 0, 0, 1, 0, 2, 0, 4, 0, 5, 0, 9, 0, 13, 0, 23, 0, 34, 0, 54, 0, 78, 0, 120, 0, 170, 0, 252, 0, 358, 0, 517, 0, 725, 0, 1030, 0, 1427, 0, 1992, 0, 2733, 0, 3759, 0, 5106, 0, 6946, 0, 9345, 0, 12577, 0, 16788, 0, 22384, 0, 29641, 0

Offset: 0

Gus Wiseman, Oct 01 2022

nonn

A partition of n is a weakly decreasing sequence of positive integers summing to n.

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

			The a(1) = 1 through a(13) = 9 partitions:
  1   .  .  .  311   .  322   .  333     .  443     .  553
                        421      432        542        652
                                 531        641        751
                                 51111      52211      52222
                                            62111      53311
                                                       62221
                                                       63211
                                                       73111
                                                       7111111

For product equal to sum we have A001055, compositions A335405.
The version for compositions is A357182, reverse ranked by A357184.
The reverse version is A357189, ranked by A357486.
These partitions are ranked by A357485.
Removing zeros gives A357488.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
Cf. A004526, A051159, A114220, A131044, A262046, A262977, A301987, A357183.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[Reverse[#]]&]],{n,0,30}]

A357185 Numbers k such that the k-th composition in standard order has the same length as the absolute value of its alternating sum.

Original entry on oeis.org

0, 1, 9, 12, 19, 22, 28, 34, 40, 69, 74, 84, 97, 104, 132, 135, 141, 144, 153, 177, 195, 198, 204, 216, 225, 240, 265, 271, 274, 283, 286, 292, 307, 310, 316, 321, 328, 355, 358, 364, 376, 386, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 544, 553

Offset: 1

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. 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 sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   12: (1,3)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   40: (2,4)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
   97: (1,5,1)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For sum equal to twice alternating sum we have A348614, counted by A262977.
For product equal to sum we have A335404, counted by A335405.
These compositions are counted by A357183.
This is the absolute value version of A357184, counted by A357183.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==Abs[ats[stc[#]]]&]

A163455 a(n) = binomial(5*n-1,n).

Original entry on oeis.org

1, 4, 36, 364, 3876, 42504, 475020, 5379616, 61523748, 708930508, 8217822536, 95722852680, 1119487075980, 13136858812224, 154603005527328, 1824010149372864, 21566576904406820, 255485622301674660, 3031718514166879020, 36030431772522503316

Offset: 0

Zak Seidov, Jul 28 2009

Also, number of terms in A163142 with n zeros in binary representation.

All terms >= 4 are divisible by 4.

			a(1)=4 because there are 4 terms in A163142 with 1 zero in binary representation {23,27,29,30}_10 ={10111,11011,11101,11110}_2
a(2)=36 because there are 36 terms in A163142 with 2 zeros in binary representation: {639,703,735,751,759,763,765,766,831,863,879,887,891,893,894,927,943,951,955,957,958,975,983,987,989,990,999,1003,1005,1006,1011,1013,1014,1017,1018,1020}_10={1001111111,...,1111111100}_2
a(3)=364 terms in A163142 from 18431 to 32760 with 3 zeros in binary representation 18431_10=100011111111111_2 and 32760_10=111111111111000_2
a(4)=3876 terms in A163142 from 557055 to 1048560 with 4 zeros in binary representation, etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A163142, A118971.

Cf. A088218, A165817, A262977.

[Binomial(5*n-1, n): n in [0..30]]; // Vincenzo Librandi, Aug 07 2014

Table[(5*n-1)!/ n!/(4*n-1)!,{n,20}]
Table[Binomial[5 n - 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

B(x):=sum(binomial(5*n-2,n-1)/(n)*x^n,n,1,30);
taylor(x*diff(B(x),x,1)/B(x),x,0,10);

a(n) = binomial(5*n-1,n); \\ Michel Marcus, Oct 06 2015

a(n) = (5n-1)!/(n!(4n-1)!).
G.f.: A(x)=x*B'(x)/B(x), where B(x)/x is g.f. for A118971. Also a(n) = Sum_{k=0..n} (binomial(n-1,n-k)*binomial(4*n,k)). - Vladimir Kruchinin, Oct 06 2015
From Peter Bala, Feb 14 2024: (Start)
a(n) = (-1)^n * binomial(-4*n, n).
a(n) = hypergeom([1 - 4*n, -n], [1], 1).
A(x) satisfies A(x/(1 + x)^5) = 1/(1 - 4*x). (End)
From Peter Bala, Jun 05 2024: (Start)
Right-hand side of the identity Sum_{k = 0..n} binomial(n+k-1, k)*binomial(4*n-k-1, n-k) = binomial(5*n-1, n).
a(n) = (3/4)*binomial(4*n, 3*n)*hypergeom([n, -n], [1 - 4*n], 1) for n >= 1. (End)
From Karol A. Penson Jan 20 2025: (Start)
G.f.: 4*z*Hypergeometric5F4([1, 6/5, 7/5, 8/5, 9/5], [5/4, 3/2, 7/4, 2], (3125*z)/256) + 1.
G.f. A(z) satisfies: z*(1250*A^3 - 250*A^2 + 25*A - 1) + (-3125*z + 256)*A^4 + (3125*z - 256)*A^5 = 0. (End)
G.f.: 1/(5-4*g) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

Entry revised by N. J. A. Sloane, Dec 07 2015

A257633 a(n) = binomial(4*n + 2,n).

Original entry on oeis.org

1, 6, 45, 364, 3060, 26334, 230230, 2035800, 18156204, 163011640, 1471442973, 13340783196, 121399651100, 1108176102180, 10142940735900, 93052749919920, 855420636763836, 7877932561061640, 72667580816130436, 671262558647881200, 6208770443303347920

Offset: 0

Peter Bala, Nov 04 2015

Michael De Vlieger, Table of n, a(n) for n = 0..1025
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Cf. A002293, A004331, A005810, A052203, A224274, A262977.

#A257633
seq(binomial(4*n + 2,n), n = 0..20);

Table[Binomial[4*n + 2, n], {n, 0, 120}] (* Michael De Vlieger, Apr 11 2025 *)

vector(30, n, n--; binomial(4*n+2, n)) \\ Altug Alkan, Nov 05 2015

The o.g.f. equals f(x)*g(x)^2, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A224274 (k = 3) and A004331 (k = 4).

A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

Original entry on oeis.org

0, 9, 130, 135, 141, 153, 177, 193, 225, 2052, 2059, 2062, 2069, 2074, 2079, 2089, 2098, 2103, 2109, 2129, 2146, 2151, 2157, 2169, 2209, 2242, 2247, 2253, 2265, 2289, 2369, 2434, 2439, 2445, 2457, 2481, 2529, 2561, 2689, 2818, 2823, 2829, 2841, 2865, 2913

Offset: 1

Gus Wiseman, Nov 22 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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The terms and corresponding compositions begin:
     0: ()
     9: (3,1)
   130: (6,2)
   135: (5,1,1,1)
   141: (4,1,2,1)
   153: (3,1,3,1)
   177: (2,1,4,1)
   193: (1,6,1)
   225: (1,1,5,1)
  2052: (9,3)
  2059: (8,2,1,1)
  2062: (8,1,1,2)
  2069: (7,2,2,1)
  2074: (7,1,2,2)
  2079: (7,1,1,1,1,1)
  2089: (6,2,3,1)
  2098: (6,1,3,2)
  2103: (6,1,2,1,1,1)

These compositions are counted by A224274 up to 0's.
An unordered version is A348617, counted by A001523 up to 0's.
The positive version is A349153, unreversed A348614.
The unreversed version is A349154.
Positive unordered unreversed: A349159, counted by A000712 up to 0's.
A positive unordered version is A349160, counted by A006330 up to 0's.
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 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, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.
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.
- Heinz number is given by A333219.
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.
- Alternating compositions are ranked by A345167, complement A345168.

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

A371771 a(n) = Sum_{k=0..floor(n/3)} binomial(4n-3k-1,n-3*k).

Original entry on oeis.org

1, 3, 21, 166, 1377, 11748, 102088, 898677, 7987305, 71517307, 644134026, 5829345492, 52964836184, 482846377185, 4414405051413, 40458397722306, 371605426607673, 3419639400458316, 31521758873514301, 291000881055737811, 2690082750919841442

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A005810, A147855, A262977.

Cf. A371758, A371770, A371772.

f:= proc(n) local k; add(binomial(4*n-3*k-1,n-3*k),k=0..n/3) end proc:
map(f, [$0..30]); # Robert Israel, Feb 28 2025

a(n) = sum(k=0, n\3, binomial(4*n-3*k-1, n-3*k));

a(n) = [x^n] 1/((1-x^3) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-4*n)/3, 2*(1-2*n)/3, 1-4*n/3], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(9037*n^4 - 61391*n^3 + 154035*n^2 - 169317*n + 68836)*a(n) = 27*(2394805*n^7 - 19820156*n^6 + 66654684*n^5 - 117198990*n^4 + 115250735*n^3 - 62650734*n^2 + 17209736*n - 1814400)*a(n-1) - 3*(7021749*n^7 - 58192764*n^6 + 196050236*n^5 - 345531070*n^4 + 340849311*n^3 - 186035886*n^2 + 51353864*n - 5443200)*a(n-2) + 8*(2*n - 3)*(4*n - 9)*(4*n - 7)*(9037*n^4 - 25243*n^3 + 24084*n^2 - 9272*n + 1200)*a(n-3).
a(n) ~ 2^(8*n + 9/2) / (7 * sqrt(Pi*n) * 3^(3*n + 3/2)). (End)

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

Original entry on oeis.org

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[#]]&]

A386701 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n,k).

Original entry on oeis.org

1, 1, 13, 103, 869, 7476, 65323, 577242, 5144949, 46167196, 416527828, 3774785983, 34336862435, 313330665532, 2866982877954, 26294890918308, 241665561294741, 2225104901535564, 20520648006149980, 189523353219338572, 1752680220372189364, 16227703263403842768

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A386700, A386702.

Cf. A066381, A262977, A371814, A385498.

Table[(-16/27)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = sum(k=0, n, (-3)^(n-k)*binomial(4*n, k));

a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(612*n^3 - 2838*n^2 + 4354*n - 2209)*a(n) = 24*(165240*n^6 - 1019628*n^5 + 2493432*n^4 - 3068178*n^3 + 1984652*n^2 - 632900*n + 76545)*a(n-1) + 128*(2*n - 3)*(4*n - 7)*(4*n - 5)*(612*n^3 - 1002*n^2 + 514*n - 81)*a(n-2).
a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). (End)
G.f.: g/((-2+3*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-8+9*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A387037 a(n) = Sum_{k=0..n} binomial(4*n-1,k).

Original entry on oeis.org

1, 4, 29, 232, 1941, 16664, 145499, 1285624, 11460949, 102875128, 928495764, 8417689504, 76599066579, 699232769512, 6400175653922, 58718827590992, 539822826733397, 4971747032359352, 45863130731297180, 423683961417124576, 3919058645835901556

Offset: 0

Seiichi Manyama, Aug 13 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387035, A387036.
Cf. A114121, A387033.
Cf. A002293, A262977.

[&+[Binomial(4*n-1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Table[Sum[Binomial[4*n-1,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)

a(n) = sum(k=0, n, binomial(4*n-1, k));

a(n) = [x^n] (1+x)^(4*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-1,k) * binomial(4*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-2,n-k).
G.f.: 1/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-5)*(4*n-7)*(2*n-3)*(11*n^2-3*n-3)*a(n-2) -8*(946*n^5-4218*n^4+6512*n^3-3753*n^2+201*n+315)*a(n-1) +3*n*(3*n-2)*(3*n-4)*(11*n^2-25*n+11)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi*n) * 3^(3*n - 3/2)). - Vaclav Kotesovec, Sep 03 2025

cp's OEIS Frontend

A357487 Number of integer partitions of n with the same length as reverse-alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A357185 Numbers k such that the k-th composition in standard order has the same length as the absolute value of its alternating sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A163455 a(n) = binomial(5*n-1,n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

Extensions

A257633 a(n) = binomial(4*n + 2,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A371771 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n-3*k-1,n-3*k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

A371771 a(n) = Sum_{k=0..floor(n/3)} binomial(4n-3k-1,n-3*k).