A014668 -id:A014668 - cp's OEIS frontend

A163313 Triangle read by rows, A010766 convolved with A014668 (diagonalized as an infinite lower triangular matrix).

1, 2, 1, 3, 1, 3, 4, 2, 3, 7, 5, 2, 3, 7, 16, 6, 3, 6, 7, 16, 33, 7, 3, 6, 7, 16, 33, 71, 8, 4, 6, 14, 16, 33, 71, 143, 9, 4, 9, 14, 16, 33, 71, 143, 295, 10, 5, 9, 14, 32, 33, 71, 143, 295, 594

Offset: 1

Gary W. Adamson & Mats Granvik, Jul 30 2009

This is an eigentriangle (i.e., a lower triangular matrix * a diagonalized version of its eigensequence); A014668 is the eigensequence of triangle A010766.
Row sums = A014668 starting (1, 3, 7, 16, 33, 71, 143, ...).
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
   1;
   2,  1;
   3,  1,  3;
   4,  2,  3,  7;
   5,  2,  3,  7, 16;
   6,  3,  6,  7, 16, 33;
   7,  3,  6,  7, 16, 33  71;
   8,  4,  6, 14, 16, 33, 71, 143;
   9,  4,  9, 14, 16, 33, 71, 143, 295;
  10,  5,  9, 14, 32, 33, 71, 143, 295, 594;
  11,  5,  9, 14, 32, 33, 71, 143, 295, 594, 1206;
  12,  6, 12, 21, 32, 66, 71, 143, 295, 594, 1206, 2413;
  ...
Example: row 4 = (4, 2, 3, 7) = (4, 2, 1, 1) * (1, 1, 3, 7).

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A010766, A014668.

a[1] = 1; a[n_] := a[n] = Sum[Sum[a[d], {d, Divisors[k]}], {k, 1, n -1}];
Table[Floor[n/k]* a[k], {n, 1, 5}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 18 2016 *)

Equals M * Q as infinite lower triangular matrices, where M = triangle A010766 and Q = a matrix with A014668: (1, 1, 3, 7, 16, 33, 71, 143, ...) as the main diagonal and the rest zeros.

A010766 Triangle read by rows: row n gives the numbers floor(n/k), k = 1..n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 5, 2, 1, 1, 1, 6, 3, 2, 1, 1, 1, 7, 3, 2, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 13, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane

Number of times k occurs as divisor of numbers not greater than n. - Reinhard Zumkeller, Mar 19 2004
Viewed as a partition, row n is the smallest partition that contains every partition of n in the usual ordering. - Franklin T. Adams-Watters, Mar 11 2006
Row sums = A006218. - Gary W. Adamson, Oct 30 2007
A014668 = eigensequence of the triangle. A163313 = A010766 * A014668 (diagonalized) as an infinite lower triangular matrix. - Gary W. Adamson, Jul 30 2009
A018805(T(n,k)) = A242114(n,k). - Reinhard Zumkeller, May 04 2014
Viewed as partitions, all rows are self-conjugate. - Matthew Vandermast, Sep 10 2014
Row n is the partition whose Young diagram is the union of Young diagrams of all partitions of n (rewording of Franklin T. Adams-Watters's comment). - Harry Richman, Jan 13 2022

			Triangle starts:
   1:  1;
   2:  2,  1;
   3:  3,  1, 1;
   4:  4,  2, 1, 1;
   5:  5,  2, 1, 1, 1;
   6:  6,  3, 2, 1, 1, 1;
   7:  7,  3, 2, 1, 1, 1, 1;
   8:  8,  4, 2, 2, 1, 1, 1, 1;
   9:  9,  4, 3, 2, 1, 1, 1, 1, 1;
  10: 10,  5, 3, 2, 2, 1, 1, 1, 1, 1;
  11: 11,  5, 3, 2, 2, 1, 1, 1, 1, 1, 1;
  12: 12,  6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
  13: 13,  6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  14: 14,  7, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  15: 15,  7, 5, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
  16: 16,  8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
  17: 17,  8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  18: 18,  9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  19: 19,  9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  20: 20, 10, 6, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ...

Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.

T. D. Noe, Rows n = 1..50 of triangle, flattened

Another version of A003988.
Finite differences of rows: A075993.
Cf. related triangles: A002260, A013942, A051731, A163313, A277646, A277647.
Cf. related sequences: A006218, A014668, A115725.
Columns of this triangle:
T(n,1) = n,
T(n,2) = A008619(n-2) for n>1,
T(n,3) = A008620(n-3) for n>2,
T(n,4) = A008621(n-4) for n>3,
T(n,5) = A002266(n) for n>4,
T(n,n) = A000012(n) = 1.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033322(k),
T(3,k) = A278105(k),
T(4,k) = A033324(k),
T(5,k) = A033325(k),
T(6,k) = A033326(k),
T(7,k) = A033327(k),
T(8,k) = A033328(k),
T(9,k) = A033329(k),
T(10,k) = A033330(k),
...
T(99,k) = A033419(k),
T(100,k) = A033420(k),
T(1000,k) = A033421(k),
T(10^4,k) = A033422(k),
T(10^5,k) = A033427(k),
T(10^6,k) = A033426(k),
T(10^7,k) = A033425(k),
T(10^8,k) = A033424(k),
T(10^9,k) = A033423(k).

a010766 = div
a010766_row n = a010766_tabl !! (n-1)
a010766_tabl = zipWith (map . div) [1..] a002260_tabl
-- Reinhard Zumkeller, Apr 29 2015, Aug 13 2013, Apr 13 2012

seq(seq(floor(n/k),k=1..n),n=1..20); # Robert Israel, Sep 01 2014

Flatten[Table[Floor[n/k],{n,20},{k,n}]] (* Harvey P. Dale, Nov 03 2012 *)

a(n)=t=floor((-1+sqrt(1+8*(n-1)))/2);(t+1)\(n-t*(t+1)/2) \\ Edward Jiang, Sep 10 2014

T(n, k) = sum(i=1, n, (i % k) == 0); \\ Michel Marcus, Apr 08 2017

G.f.: 1/(1-x)*Sum_{k>=1} x^k/(1-y*x^k). - Vladeta Jovovic, Feb 05 2004
Triangle A010766 = A000012 * A051731 as infinite lower triangular matrices. - Gary W. Adamson, Oct 30 2007
Equals A000012 * A051731 as infinite lower triangular matrices. - Gary W. Adamson, Nov 14 2007
Let T(n,0) = n+1, then T(n,k) = (sum of the k preceding elements in the previous column) minus (sum of the k preceding elements in same column). - Mats Granvik, Gary W. Adamson, Feb 20 2010
T(n,k) = (n - A048158(n,k)) / k. - Reinhard Zumkeller, Aug 13 2013
T(n,k) = 1 + T(n-k,k) (where T(n-k,k) = 0 if n < 2*k). - Robert Israel, Sep 01 2014
T(n,k) = T(floor(n/k),1) if k>1; T(n,1) = 1 - Sum_{i=2..n} A008683(i)*T(n,i). If we modify the formula to T(n,1) = 1 - Sum_{i=2..n} A008683(i)*T(n,i)/i^s, where s is a complex variable, then the first column becomes the partial sums of the Riemann zeta function. - Mats Granvik, Apr 27 2016

Cross references edited by Jason Kimberley, Nov 23 2016

A320222 Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 39, 78, 161, 324, 658, 1316, 2657, 5314, 10668, 21347, 42777, 85554, 171290, 342580, 685498, 1371037, 2742733, 5485466, 10972351, 21944711, 43892080, 87784323, 175574004, 351148008, 702307038, 1404614076, 2809249582, 5618499824, 11237042426

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

This is a weaker condition than achirality (cf. A003238).

			The a(1) = 1 through a(6) = 18 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o)(o))   (((ooo)))
                          ((o(o)))   ((o(oo)))
                          (o((o)))   ((oo(o)))
                          ((((o))))  (o((oo)))
                                     (o(o)(o))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002541, A003238, A010766, A126656, A014668, A167865, A214577, A316782, A317099, A317100, A317712, A320230.

saue[n_]:=Sum[If[SameQ@@DeleteCases[ptn,1],If[DeleteCases[ptn,1]=={},1,saue[DeleteCases[ptn,1][[1]]]],0],{ptn,IntegerPartitions[n-1]}];
Table[saue[n],{n,15}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=2, n-1, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(n) = 1 + Sum_{k = 2..n-1} floor((n-1)/k) * a(k).

a(n) ~ c * 2^n, where c = 0.3270422384018894564479397100499014525700668391191792769625407295138546463... - Vaclav Kotesovec, Sep 07 2019

A126656 a(1)=1; for n>1, a(n) = Sum_{k=1..n-1} a(k) * floor(n/k).

Original entry on oeis.org

1, 2, 5, 13, 27, 62, 125, 266, 538, 1106, 2213, 4509, 9019, 18166, 36365, 73012, 146025, 292658, 585317, 1171783, 2343697, 4689610, 9379221, 18763300, 37526628, 75062278, 150125100, 300268507, 600537015, 1201111598, 2402223197

Offset: 1

Leroy Quet, Feb 09 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..3000 (terms 1..333 from Robert Israel)

Cf. A014668, A320225.

sol:=[1]; for n in [2..31] do Append(~sol, &+[sol[k]*Floor(n/k):k in [1..n-1]]); end for; sol; // Marius A. Burtea, Sep 07 2019

A126656[1]:= 1:
for n from 2 to 100 do
  A126656[n]:= add(A126656[k]*floor(n/k),k=1..n-1);
od:
[seq(A126656[i],i=1..100)]; # Robert Israel, May 11 2014

f[l_List] := Block[{n = Length[l] + 1},Append[l, Sum[l[[k]]*Floor[n/k], {k, n - 1}]]];Nest[f, {1}, 32] (* Ray Chandler, Feb 11 2007 *)

a(n) is asymptotic to c*2^n where c=1.1186567921399193608303756752855835542518061417771006467002421299046146... [Benoit Cloitre, Nov 10 2009]

Extended by Ray Chandler, Feb 11 2007

A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 44, 70, 119, 189, 314, 506, 830, 1336, 2186, 3522, 5737, 9266, 15047, 24313, 39444, 63759, 103322, 167098, 270616, 437714, 708676, 1146390, 1855582, 3002017, 4858429, 7860454, 12720310, 20580764, 33303260, 53884144, 87190964

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

This is a weaker condition than achirality (cf. A167865).

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 1 through a(8) = 9 rooted trees:
  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
               (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
                        (oo(oo))  (oo(ooo))   (oo(oooo))
                                  (ooo(oo))   (ooo(ooo))
                                  ((oo)(oo))  (oooo(oo))
                                  (o(o(oo)))  (o(o(ooo)))
                                              (o(oo)(oo))
                                              (o(oo(oo)))
                                              (oo(o(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001678, A002541, A003238, A010766, A070776, A014668, A126656, A167865, A317099, A317100, A317712, A320222, A320226, A320269.

saum[n_]:=Sum[If[DeleteCases[ptn,1]=={},1,saum[DeleteCases[ptn,1][[1]]]],{ptn,Select[IntegerPartitions[n-1],And[Length[#]!=1,SameQ@@DeleteCases[#,1]]&]}];
Array[saum,20]

seq(n)={my(v=vector(n)); v[1]=1; for(n=3, n, v[n] = 1 + sum(k=2, n-2, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(1) = 1; a(2) = 0; a(n > 2) = 1 + Sum_{k = 2..n-2} floor((n-1)/k) * a(k).

A320224 a(1) = 1; a(n > 1) = Sum_{k = 1..n-1} Sum_{d|k, d < k} a(d).

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 7, 10, 12, 16, 17, 25, 26, 33, 38, 48, 49, 65, 66, 84, 92, 109, 110, 142, 146, 172, 184, 219, 220, 274, 275, 323, 341, 390, 400, 484, 485, 551, 578, 669, 670, 792, 793, 904, 952, 1062, 1063, 1243, 1250, 1408, 1458, 1632, 1633, 1870, 1890

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002541, A003238, A010766, A014668, A126656, A167865, A316782, A317712, A320222, A320225.

sol:=[1]; for n in [2..56] do Append(~sol, &+[sol[d]*Floor((n-1)/d-1):d in [1..n-1]]); end for; sol; // Marius A. Burtea, Sep 07 2019

sau[n_]:=If[n==1,1,Sum[sau[d],{k,n-1},{d,Most[Divisors[k]]}]];
Table[sau[n],{n,60}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(k=1, n-1, v[k]*((n-1)\k - 1))); v} \\ Andrew Howroyd, Sep 07 2019

a(1) = 1; a(n > 1) = Sum_{d = 1..n-1} a(d) * floor((n-1)/d - 1).

G.f. A(x) satisfies A(x) = x + (x/(1 - x)) * Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019

A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

Original entry on oeis.org

1, 1, 2, 4, 5, 9, 10, 16, 19, 26, 27, 44, 45, 57, 65, 87, 88, 120, 121, 158, 171, 200, 201, 278, 284, 331, 353, 426, 427, 536, 537, 646, 676, 766, 782, 982, 983, 1106, 1154, 1365, 1366, 1617, 1618, 1851, 1943, 2146, 2147, 2589, 2600, 2917, 3008, 3390, 3391

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002541, A003238, A010766, A014668, A126656, A167865, A316782, A317712, A320222, A320224.

sau[n_]:=If[n==1,1,Sum[sau[d],{k,n},{d,Most[Divisors[k]]}]];
Table[sau[n],{n,30}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A320225(n): return 1 if n == 1 else sum(A320225(d)*(n//d-1) for d in range(1,n)) # Chai Wah Wu, Jun 08 2022

a(1) = 1; a(n > 1) = Sum_{d = 1..n-1} a(d) * floor(n/d-1).

G.f. A(x) satisfies A(x) = x + (1/(1 - x)) * Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019

A332846 a(1) = 1; a(n+1) = Sum_{k=1..n} a(k) * ceiling(n/k).

Original entry on oeis.org

1, 1, 3, 8, 20, 50, 121, 297, 716, 1739, 4198, 10157, 24513, 59246, 143006, 345381, 833792, 2013272, 4860337, 11734717, 28329772, 68396030, 165121957, 398644144, 962410246, 2323475153, 5609360573, 13542220814, 32693802921, 78929886033, 190553574988, 460037180829, 1110627936647

Offset: 1

Ilya Gutkovskiy, Feb 26 2020

nonn

Cf. A006590, A014668, A097919, A332490.

a[1] = 1; a[n_] := a[n] = Sum[a[k] Ceiling[(n - 1)/k], {k, 1, n - 1}]; Table[a[n], {n, 1, 33}]
a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] + Sum[a[d], {d, Divisors[k]}], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
terms = 33; A[] = 0; Do[A[x] = x (1 + (1/(1 - x)) (A[x] + x Sum[A[x^k], {k, 1, terms}])) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * (A(x) + x * Sum_{k>=1} A(x^k))).
a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-2} (a(k) + Sum_{d|k} a(d)).
a(n) ~ c * (1 + sqrt(2))^n, where c = 0.2594006517235012546870541901936538347053403598092060748627156661727... - Vaclav Kotesovec, Mar 10 2020

A333494 a(1) = 1; a(n) = Sum_{k=1..n-1} ceiling(n/k) * a(k).

Original entry on oeis.org

1, 2, 7, 22, 69, 208, 634, 1903, 5734, 17210, 51702, 155107, 465561, 1396684, 4190689, 12572144, 37718360, 113155081, 339471195, 1018413586, 3055258062, 9165774828, 27497376189, 82492128568, 247476542954, 742429628932, 2227289352360, 6681868062822, 20045605585809

Offset: 1

Ilya Gutkovskiy, Mar 24 2020

nonn

Cf. A006590, A014668, A126656, A332846.

a[1] = 1; a[n_] := a[n] = Sum[Ceiling[n/k] a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 29}]
terms = 29; A[] = 0; Do[A[x] = x (1 + (1/(1 - x)) (A[x] + Sum[A[x^k], {k, 1, terms}])) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * (A(x) + Sum_{k>=1} A(x^k))).

a(n) ~ c * 3^n, where c = 0.292080665386646518390576592052254840432101999262173908555857806023213143845... - Vaclav Kotesovec, Mar 25 2020

cp's OEIS Frontend

A163313 Triangle read by rows, A010766 convolved with A014668 (diagonalized as an infinite lower triangular matrix).

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A010766 Triangle read by rows: row n gives the numbers floor(n/k), k = 1..n.

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A320222 Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.

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A126656 a(1)=1; for n>1, a(n) = Sum_{k=1..n-1} a(k) * floor(n/k).

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A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

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A320224 a(1) = 1; a(n > 1) = Sum_{k = 1..n-1} Sum_{d|k, d < k} a(d).

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A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

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