A028361 -id:A028361 - cp's OEIS frontend

A131791 Triangle read by rows of 2^n terms for n>=0: let S(n) denote the initial 2^n terms of the partial sums of row n; starting with a single '1' in row 0, generate row n+1 by concatenating S(n) with the terms of S(n) when read in reverse order.

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 6, 5, 3, 1, 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1, 1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269, 270, 270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14, 5, 1, 1, 6, 20, 49, 99, 175, 280, 415

Offset: 0

Paul D. Hanna, Jul 15 2007

Row sums (and central terms) form A028361: Product_{i=0..n-1} (2^i + 1).
I'm interested in the graph of S(n). It appears to tend to a limit curve if scaled appropriately, e.g., scaled to fit a [0,1] box by f_n(x) = T(n,[x*2^n])/A028361(n-1). In this setup I think that the limit curve f(x) satisfies f(0)=0, f(1-x)=f(x), f(1/2)=1, f'(x)=2f(2x) for x<=1/2. Is this equation solvable? - Martin Fuller, Aug 31 2007
From N. J. A. Sloane, Nov 13 2018: (Start)
Kenyon (1992) defines p_n(x) (n >= 0) to be the polynomial
p_n(x) = (1+x)*(1+x+x^2)*(1+x+x^2+x^3+x^4)*...*(1+x+...+x^(2^n)).
He shows among many other things that the coefficient of x^(floor(c*2^(n+1))) in p_n(x), for c in [0,1], is given by
(f(c)+o(1))*p_n(1)/2^(n+1),
where f : R -> R is a nonzero C^1 function satisfying
(i) support(f) is a subset of [0,1],
(ii) f(x) = f(1-x), and
(iii) f'(x) = 4*f(2*x) for 0 <= x <= 1/2.
These three properties define f uniquely up to multiplication by a scalar. Also f is C^oo, is nowhere analytic on [0,1], and is a "bump function", since its graph looks like a "bump".
This provides a fairly complete answer to Martin Fuller's question above. (End)

			Triangle begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 5, 6, 6, 5, 3, 1;
1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1;
1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269, 270, 270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14, 5, 1; ...
ILLUSTRATION OF GENERATING METHOD.
From row 2: [1,2,2,1], take the partial sums: [1,3,5,6] and concatenate to this the terms in reverse order: [6,5,3,1] to obtain row 3: [1,3,5,6, 6,5,3,1].

Richard Kenyon, Infinite scaled convolutions, Preprint, 1992 (apparently unpublished)

Paul D. Hanna, Rows 0 to 11 of the triangle, flattened.
Julien Clément, Antoine Genitrini, Binary Decision Diagrams: from Tree Compaction to Sampling, arXiv:1907.06743 [cs.DS], 2019.

Cf. A131792 (main diagonal); A028361, A028362.

p[-1]:=1:
lprint(seriestolist(series(p[-1],x,0)));
p[0]:=(1-x^2)/(1-x):
lprint(seriestolist(series(p[0],x,2)));
for n from 1 to 4 do
p[n]:=p[n-1]*(1-x^(2^n+1))/(1-x);
lprint(seriestolist(series(p[n],x,2^(n+1))));
od: # N. J. A. Sloane, Nov 13 2018

T[n_, k_] := SeriesCoefficient[Product[(1-x^(2^j+1))/(1-x), {j, 0, n-1}], {x, 0, k}];
Table[T[n, k], {n, 0, 6}, {k, 0, 2^n-1}] // Flatten (* Jean-François Alcover, Oct 01 2019 *)

T(n,k)=local(A=[1],B=[1]);if(n==0,1,for(i=0,n-1, B=Vec(Ser(A)/(1-x));A=concat(B,Vec(Pol(B)+O(x^#B))));A[k+1])
for(n=0,6,for(k=0,2^n-1,print1(T(n,k),", "));print())

T(n,k)=polcoeff(prod(j=0,n-1,(1-x^(2^j+1))/(1-x)),k)
for(n=0,6,for(k=0,2^n-1,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Aug 09 2009

T(n, 2^(n-1)) = A028361(n-1) for n>=1.
T(n, 2^(n-2)) = A028362(n-1) for n>=2.
Sum_{k=0..2^n-1} (k+1)*T(n,k) = A028362(n+1) for n>=0.
G.f. of row n: Product_{j=0..n-1} (1 - x^(2^j+1))/(1-x). - Paul D. Hanna, Aug 09 2009

A139486 a(n) = Product_{j=0..n-1} (2^j + 2).

Original entry on oeis.org

1, 3, 12, 72, 720, 12960, 440640, 29082240, 3780691200, 975418329600, 501365021414400, 514400511971174400, 1054521049540907520000, 4321427261018639016960000, 35409774976786728104970240000, 580224572769627326728042352640000

Offset: 0

Leroy Quet, Apr 23 2008

Cf. A139485, A028361.

PARI
```
{ a(n) = prod(j=0,n-1,2^j+2) }
```

a(n) = A139485(2^n + 1) - 1 = Sum_{j=1..2^n} A139485(j).

For n >= 1, a(n) = 2^(n-1) * 3 * A028361(n-1).

Edited and extended by Max Alekseyev, Oct 24 2008

A131792 Main diagonal of triangle A131791: a(n) = A131791(n,n) for n>=0.

Original entry on oeis.org

1, 1, 2, 6, 21, 76, 280, 1045, 3936, 14925, 56892, 217791, 836706, 3224157, 12456225, 48232162, 187131664, 727309265, 2831193004, 11036424667, 43076087806, 168322335246, 658416150496, 2577945422410, 10102468839284, 39621592646545

Offset: 0

Paul D. Hanna, Jul 15 2007

nonn

Row n of triangle A131791 has 2^n terms for n>=0, where row sums and central terms of A131791 equals A028361: Product_{i=0..n-1} (2^i + 1).

{a(n)=if(n==0,1,polcoeff(prod(j=0,n-1,(1-x^(2^j+1))/(1-x)+x*O(x^n)),n))} \\ Max Alekseyev, Aug 30 2007

{T(n,k)=if(n==0,1,polcoeff(prod(j=0,n-1,(1-x^min(2^j+1,k+1))/(1-x)+x*O(x^k)),k))} \\ Martin Fuller, Aug 31 2007

a(n) = [x^n] Product_{j=0..n-1} [ (1 - x^(2^j+1) ) / (1-x) ] for n>0, with a(0)=1. - Max Alekseyev, Aug 30 2007.

More terms from Max Alekseyev, Aug 30 2007.

A180174 Triangle read by rows of the numbers C(n,k) of k-subsets of a quadratically populated n-multiset M.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 5, 7, 9, 10, 10, 10, 10, 10, 9, 7, 5, 3, 1, 1, 4, 9, 16, 25, 35, 45, 55, 65, 75, 84, 91, 96, 99, 100, 100, 100, 99, 96, 91, 84, 75, 65, 55, 45, 35, 25, 16, 9, 4, 1, 1, 5, 14, 30, 55, 90, 135, 190, 255, 330, 414, 505, 601, 700, 800, 900, 1000, 1099

Offset: 0

Thomas Wieder, Aug 15 2010

The multiplicity m(i) of the i-th element with 1 <= i <= n is m(i)=i^2.
Thus M=[1,2,2,2,2,...,i^2 x i,...,n^2 x n].
Row sum is equal to A028361.
Column for k=2 is equal to AA000096.
Column for k=3 is equal to AA005581.
Column for k=4 is equal to AA005582.
The number of coefficients C(n,k) for given n is equal to A056520.

			For n=4 one has M=[1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4].
For k=7 we have 55 subsets from M:
[1, 2, 2, 3, 3, 4, 4], [1, 2, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 4, 4],
[1, 2, 2, 3, 4, 4, 4], [1, 2, 2, 3, 3, 3, 4], [1, 2, 2, 2, 3, 4, 4],
[1, 2, 2, 2, 3, 3, 4], [2, 2, 3, 3, 4, 4, 4], [2, 2, 3, 3, 3, 4, 4],
[2, 2, 2, 3, 3, 4, 4], [1, 2, 2, 2, 3, 3, 3], [1, 2, 2, 2, 4, 4, 4],
[1, 3, 3, 3, 4, 4, 4], [2, 3, 3, 3, 4, 4, 4], [2, 2, 2, 3, 4, 4, 4],
[2, 2, 2, 3, 3, 3, 4], [1, 2, 3, 4, 4, 4, 4], [1, 2, 3, 3, 3, 3, 4],
[1, 2, 2, 2, 2, 3, 4], [1, 2, 2, 3, 3, 3, 3], [1, 2, 2, 2, 2, 3, 3],
[1, 2, 2, 4, 4, 4, 4], [1, 2, 2, 2, 2, 4, 4], [1, 3, 3, 4, 4, 4, 4],
[1, 3, 3, 3, 3, 4, 4], [2, 3, 3, 4, 4, 4, 4], [2, 3, 3, 3, 3, 4, 4],
[2, 2, 3, 4, 4, 4, 4], [2, 2, 3, 3, 3, 3, 4], [2, 2, 2, 2, 3, 4, 4],
[2, 2, 2, 2, 3, 3, 4], [2, 2, 2, 3, 3, 3, 3], [2, 2, 2, 2, 3, 3, 3],
[2, 2, 2, 4, 4, 4, 4], [2, 2, 2, 2, 4, 4, 4], [3, 3, 3, 4, 4, 4, 4],
[3, 3, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 3, 3], [1, 2, 4, 4, 4, 4, 4],
[1, 3, 4, 4, 4, 4, 4], [1, 3, 3, 3, 3, 3, 4], [2, 3, 4, 4, 4, 4, 4],
[2, 3, 3, 3, 3, 3, 4], [2, 2, 3, 3, 3, 3, 3], [2, 2, 4, 4, 4, 4, 4],
[3, 3, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 4, 4], [1, 3, 3, 3, 3, 3, 3],
[1, 4, 4, 4, 4, 4, 4], [2, 3, 3, 3, 3, 3, 3], [2, 4, 4, 4, 4, 4, 4],
[3, 4, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 3, 4], [3, 3, 3, 3, 3, 3, 3],
[4, 4, 4, 4, 4, 4, 4].

Cf. A007318, A008302, A028361, A056520, A000096, A005581, A005582.

with(combinat)
kend := 4;
Liste := NULL;
for k from 0 to kend do
Liste := Liste, `$`(k, k^2)
end do;
Liste := [Liste];
for k from 0 to 2^(kend+1)-1 do
Teilergebnis[k] := choose(Liste, k)
end do;
seq(nops(Teilergebnis[k]), k = 0 .. 2^(kend+1)-1)
' Excel VBA
Sub A180174()
Dim n As Long, nend As Long, k As Long, kk As Long, length_row As Long, length_sum As Long
Dim ATable(10, -1000 To 1000) As Double, Summe As Double
Dim offset_row As Integer, offset_column As Integer
Worksheets("Tabelle2").Select
Cells.Select
Selection.ClearContents
Range("A1").Select
offset_row = 1
offset_column = 1
nend = 7
ATable(0, 0) = 1
Cells(0 + offset_row, 0 + offset_column) = 1
For n = 1 To nend
length_row = n * (n + 1) * (2 * n + 1) / 6
length_sum = n ^ 2 + 1
For k = 0 To length_row / 2
Summe = 0
For kk = k - length_sum + 1 To k
Summe = Summe + ATable(n - 1, kk)
Next kk
ATable(n, k) = Summe
Cells(n + offset_row, k + offset_column) = ATable(n, k)
ATable(n, length_row - k) = Summe
Cells(n + offset_row, length_row - k + 0 + offset_column) = ATable(n, k)
Next k
Next n
End Sub

C(0,0) = 0.
C(n,k) = sum_{j=(k-LS+1)}^{k} C(n-1,j).
for n > 0 and k=1,...,LR with LS = n^2+1 and LR = n*(n+1)*(2*n+1)/6.
C(n,k) = C(n,LR-k).

A260231 a(n) = Product_{k=1..n} (1 + k^k).

Original entry on oeis.org

2, 10, 280, 71960, 224946960, 10495350312720, 8643382777938679680, 145011908479540041684850560, 56180584638978557924165229531974400, 561805846445966163880630853243909229531974400, 160289764609087349005207761687490741791453382934816332800

Offset: 1

Vaclav Kotesovec, Jul 20 2015

nonn

Cf. A000312, A002109, A028361, A054640, A082480, A217757, A258325.

Table[Product[1+k^k,{k,1,n}],{n,1,12}]
FoldList[Times,Table[1+k^k,{k,12}]] (* Harvey P. Dale, Jul 19 2025 *)

a(n) ~ c * A002109(n), where c = Product_{k>=1} (1 + 1/k^k) = 2.60361190459951423330221282635022049352582879064202503882732200701325334...

A375044 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = 2^n x and t(x) = x+1. See Comments.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 10, 31, 30, 1, 19, 121, 309, 270, 1, 36, 444, 2366, 5523, 4590, 1, 69, 1632, 17018, 83601, 186849, 151470, 1, 134, 6117, 123098, 1189771, 5620914, 12296655, 9845550, 1, 263, 23403, 912191, 17069413, 159101373, 737394561, 1596114045

Offset: 1

Clark Kimberling, Sep 15 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
1 + 2x,
1 + 5 x + 6 x^2,
1 + 10 x + 31 x^2 + 30 x^3.
First 5 rows of array:
1    2
1    5     6
1   10    31    30
1   19   121   309    270
1   36   444  2366   5523   4590

Cf. A000290, A028361 (T(n,n+1)), A374848.

s[n_] := 2^n  x; t[n_] := x + 1;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A375045 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = 2^n and t(x) = 2x+1. See Comments.

Original entry on oeis.org

2, 2, 6, 10, 4, 30, 62, 40, 8, 270, 618, 484, 152, 16, 4590, 11046, 9464, 3552, 576, 32, 151470, 373698, 334404, 136144, 26112, 2208, 64, 9845550, 24593310, 22483656, 9518168, 1969568, 195744, 8576, 128, 1270075950, 3192228090, 2949578244, 1272810984

Offset: 1

Clark Kimberling, Sep 15 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
2 + 2x,
6 + 10 x + 4 x^2,
30 + 62 x + 40 x^2 + 8 x^3.
First 5 rows of array:
   2      2
   6     10     4
  30     62    40     8
 270    618   484   152   16
4590  11046  9464  3552  576  32

Cf. A000079, A028361 (column 2), A000079 (T(n,n+1)), A374848.

s[n_] := 2^n  x; t[n_] := 2x + 1;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A122579 Integer sequence arising from the nested radical R(m) = sqrt(4^0+sqrt(4^1+sqrt(....+sqrt(4^m)...))) (see comment).

Original entry on oeis.org

1, 1, 3, 15, 455, 53175, 52802967, 194817927511, 3103080851583191, 200041552904055799767, 52027813627693614618566103, 54341050032531342541499355961815

Offset: 0

Benoit Cloitre, Oct 29 2006

Let p(k) = Product_{i=0..k} (2^i+1) (A028361) then for any n>0 we get the asymptotic formula: lim_{m->oo} 2-R(m) = a(0)*(1/p(m))+a(1)*(2/p(m+1))+...+a(n)*(2^n/p(m+n))+O(1/p(m+n+1)). In particular there is this unexpected relation between an infinite nested radical and an infinite product: lim_{m->oo} (2-R(m))*2^(m*(m+1)/2) = 1/Product_{i>=0} (1+1/2^i) = 0.209... (A083864).

B. Cloitre, On an asymptotic formula for a nested radical, in preparation 2006

Cf. A028361, A083864.

a(n) is asymptotic to c*4^(n*(n-1)/2) for c=0.041....

A126391 a(1)=1; for n>1: a(n) = sum of all subsets of (a(1),..,a(n-1)).

Original entry on oeis.org

1, 1, 4, 24, 240, 4320, 146880, 9694080, 1260230400, 325139443200, 167121673804800, 171466837323724800, 351507016513635840000, 1440475753672879672320000, 11803258325595576034990080000

Offset: 1

Zak Seidov, Mar 23 2007

nonn

Is connection with A006088 clear?

Apparently first differences of A028361. - Sean A. Irvine, Jun 25 2022

			n=2: subsets of (1) are ((0),(1)), sums of subsets are (0,1) and total sum is 0+1=1, hence a(2)=1;
n=3: subsets of (1,1) are ((0),(1),(1),(1,1)), sums of subsets are (0,1,1,2) and total sum is 0+1+1+2=4, hence a(3)=4;
n=4: subsets of (1,1,4) are ((0),(1),(1),(4),(1,1),(1,4),(1,4),(1,1,4)), sums of subsets are (0,1,1,4,2,5,5,6) and total sum is 0+1+1+4+2+5+5+6=24, hence a(4)=24.

Cf. A006088.

a[1]=1;a[2]=1;a[n_]:=a[n]=(2^(n-2)+2)*a[n-1];Table[a[i],{i,18}]

a(1)=1, a(2)=1; n>2: a(n)=(2^(n-2)+2)*a(n-1). a(1)=1; n>1: a(n)=A006088(n-1).

A156895 Primes of the form 1 + Product_{j=0..k} (2^j + 1).

Original entry on oeis.org

3, 7, 31, 271, 4591, 151471, 9845551, 326409519151, 16455349947998422816223361143099916435126171767057238357700938736968751

Offset: 1

J. M. Bergot, Feb 17 2009

nonn

10th term has 150 digits. - Emeric Deutsch, Mar 07 2009

Primes of the form 1 + A028361(k), any k > 0. - R. J. Mathar, Jul 10 2011

			(2^0 + 1)*(2^1 + 1)*(2^2 + 1)*(2^3 + 1)*(2^4 + 1) + 1 = 2*3*5*9*17 + 1 = 4591.

Cf. A028361.

a := proc (n) if isprime(1+product(2^k+1, k = 0 .. n)) = true then 1+product(2^k+1, k = 0 .. n) else end if end proc: seq(a(n), n = 0 .. 40); # Emeric Deutsch, Mar 07 2009

Select[Table[Product[2^j+1,{j,0,k}],{k,0,30}]+1,PrimeQ] (* Harvey P. Dale, Aug 01 2019 *)

a(9) from Emeric Deutsch, Mar 07 2009

cp's OEIS Frontend

A131791 Triangle read by rows of 2^n terms for n>=0: let S(n) denote the initial 2^n terms of the partial sums of row n; starting with a single '1' in row 0, generate row n+1 by concatenating S(n) with the terms of S(n) when read in reverse order.

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A139486 a(n) = Product_{j=0..n-1} (2^j + 2).

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A131792 Main diagonal of triangle A131791: a(n) = A131791(n,n) for n>=0.

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A180174 Triangle read by rows of the numbers C(n,k) of k-subsets of a quadratically populated n-multiset M.

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Maple

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A260231 a(n) = Product_{k=1..n} (1 + k^k).

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A375044 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = 2^n x and t(x) = x+1. See Comments.

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Mathematica

A375045 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = 2^n and t(x) = 2x+1. See Comments.

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A122579 Integer sequence arising from the nested radical R(m) = sqrt(4^0+sqrt(4^1+sqrt(....+sqrt(4^m)...))) (see comment).

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A126391 a(1)=1; for n>1: a(n) = sum of all subsets of (a(1),..,a(n-1)).