A003506 -id:A003506 - cp's OEIS frontend

A171631 Triangle read by rows: T(n,k) = n(binomial(n-2, k-1) + nbinomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

1, 4, 2, 9, 12, 3, 16, 36, 24, 4, 25, 80, 90, 40, 5, 36, 150, 240, 180, 60, 6, 49, 252, 525, 560, 315, 84, 7, 64, 392, 1008, 1400, 1120, 504, 112, 8, 81, 576, 1764, 3024, 3150, 2016, 756, 144, 9, 100, 810, 2880, 5880, 7560, 6300, 3360, 1080, 180, 10, 121, 1100

Offset: 1

Roger L. Bagula, Dec 13 2009

If T(0,0) = 0 is prepended, then row sums give A001788.

			Triangle begins:
n\k|  0    1     2     3     4     6    7    8  9
-------------------------------------------------
1  |  1
2  |  4    2
3  |  9   12     3
4  | 16   36    24     4
5  | 25   80    90    40     5
6  | 36  150   240   180    60     6
7  | 49  252   525   560   315    84    7
8  | 64  392  1008  1400  1120   504  112    8
9  | 81  576  1764  3024  3150  2016  756  144  9
... reformatted. - _Franck Maminirina Ramaharo_, Oct 02 2018

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, 1945, p. 32.

Cf. A003506, A007318, A127952, A171531.

Table[CoefficientList[n*(x + n)*(x + 1)^(n - 2), x], {n, 1, 12}]//Flatten

T(n, k) := n*(binomial(n - 2, k - 1) + n*binomial(n - 2, k))$
tabl(nn) := for n:1 thru nn do print(makelist(T(n, k), k, 0, n - 1))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */

Let p(x;n) = (x + 1)^n. Then row n are the coefficients in the expansion of p''(x;n) - x*p'(x;n) + n*p(x;n) = n*(x + n)*(x + 1)^(n - 2).
From Franck Maminirina Ramaharo, Oct 02 2018: (Start)
T(n,1) = A000290(n).
T(n,2) = A011379(n).
T(n,3) = 3*A002417(n-2).
T(n,n-2) = A046092(n-1).
T(n,n-3) = 9*A000292(n-2).
G.f.: y*(x*y - y - 1)/(x*y + y - 1)^3. (End)

Edited and new name by Franck Maminirina Ramaharo, Oct 02 2018

A229253 Total number of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 4, 4, 12, 4, 32, 4, 32, 12, 32, 4, 192, 4, 32, 32, 80, 4, 192, 4, 192, 32, 32, 4, 1024, 12, 32, 32, 192, 4, 1024, 4, 192, 32, 32, 32, 2304, 4, 32, 32, 1024, 4, 1024, 4, 192, 192, 32, 4, 5120, 12, 192, 32, 192, 4, 1024, 32, 1024, 32, 32, 4, 24576, 4, 32, 192

Offset: 1

Jaroslav Krizek, Sep 29 2013

Number of nonempty subsets of divisors of n = A100587(n).

			For n = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; total number  of elements of subsets = 1 + 1 + 1 + 2 + 2 + 2 + 3 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000337, A001787, A003506, A100587.

with(numtheory): A229253:=n->tau(n)*2^(tau(n)-1): seq(A229253(n), n=1..100); # Wesley Ivan Hurt, Dec 12 2015

Table[Length[Flatten[Subsets[Divisors[n]]]], {n, 100}] (* T. D. Noe, Oct 01 2013 *)

A229253(n) = numdiv(n) * 2^(numdiv(n)-1); \\ Antti Karttunen, May 25 2017

a(n) = A001787(A000005(n)) = A000005(n) * 2^(A000005(n)-1) = A100587(n) + A000337(n-1) = tau(n) * 2^(tau(n)-1).

A237425 Denominators of A164555(n)/A027642(n) + A198631(n)/A006519(n+1).

Original entry on oeis.org

1, 1, 6, 4, 30, 2, 42, 8, 30, 2, 66, 4, 2730, 2, 6, 16, 510, 2, 798, 4, 330, 2, 138, 8, 2730, 2, 6, 4, 870, 2, 14322, 32, 510, 2, 6, 4, 1919190, 2, 6, 8, 13530, 2, 1806, 4, 690, 2, 282, 16, 46410, 2, 66, 4, 1590, 2, 798, 8

Offset: 0

Paul Curtz, Feb 07 2014

nonn

An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. There are two possibilities. For the first kind, the main diagonal is 0's=A000004, the first two following diagonals being the same (generally not A000004). Integers example: A000045(n).
For the second kind, the main diagonal is the double of the following diagonal. Example: the companion to A000045(n) is A000032(n)=2, 1, 3, ... .
A000032(n)/2 is also a possibility. Here a(n) is the denominator of the sum of two autosequences of second kind involving (fractional) Euler and Bernoulli numbers. The corresponding fractional sequence is also an autosequence of the second kind: 2, 1, 1/6, -1/4, -1/30, 1/2, 1/42, -17/8, -1/30, 31/2, 5/66, -691/4, -691/2730,... . It could be divided by 2.

Cf. 1/n in A003506, A194767, A218289, A168516/A168426, A224270/A046161.

a[n_] := BernoulliB[n] + EulerE[n, 1]/2^IntegerExponent[n, 2]; a[0] = 2; a[1] = 1; Table[a[n] // Denominator, {n, 0, 55}] (* Jean-François Alcover, Feb 11 2014 *)

a(2n) = A002445(n). a(2n+2) = A171977(n+2).

A276670 Numerator of (n-1)n(n+1)/4.

Original entry on oeis.org

0, 0, 3, 6, 15, 30, 105, 84, 126, 180, 495, 330, 429, 546, 1365, 840, 1020, 1224, 2907, 1710, 1995, 2310, 5313, 3036, 3450, 3900, 8775, 4914, 5481, 6090, 13485, 7440, 8184, 8976, 19635, 10710, 11655, 12654, 27417, 14820, 15990, 17220, 37023, 19866, 21285

Offset: 0

Paul Curtz, Oct 05 2016

Consider the sequence [2/(n+1), autosequence of the second kind] (see A003506), and its successive differences:
    2,    1,  2/3,   1/2,   2/5,   1/3,   2/7,   1/4,   2/9, ... (see A026741)
   -1, -1/3, -1/6, -1/10, -1/15, -1/21, -1/28, -1/36, -1/45, ... (see A000217)
  2/3,  1/6, 1/15,  1/30, 2/105,  1/84, 1/126, 1/180, 2/495, ...
  ...
Each fraction in the third row is essentially the reciprocal of (n-1)*n*(n+1)/4 (3/2, 6, 15, 30, 105/2, ... ).
The numbers (= 3*A138190) are divisible by
1) -1, 1, 1, 1, 3, 2, 5, 3, 7, ... hence f(n) = 0, 0, 3, 6, 5, 15, 21, 28, 18, ...
2)  1, 1, 3, 3, 5, 5, 7, 7, 9, ... hence g(n) = 0, 0, 1, 2, 3, 6, 15, 12, 14, ...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Cf. A000217, A003506, A007531, A027641, A138190, A109613, A236398 (denominators).

seq(numer((n^3-n)/4), n=0..100); # Robert Israel, Oct 05 2016

f[n_] := Numerator[(n - 1) n (n + 1)/4]; Array[f, 40, 0] (* Robert G. Wilson v, Oct 05 2016 *)

concat(vector(2), Vec(3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4) + O(x^30))) \\ Colin Barker, Oct 09 2016

a(n) = 3*A138190(n), for n>=1.
a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
a(n) = A007531(n+1)/2 if n == 2 (mod 4), otherwise a(n) = A007531(n+1)/4. - Robert Israel, Oct 05 2016
G.f.: 3*x^2*(1 +2*x +5*x^2 +10*x^3 +31*x^4 +20*x^5 +22*x^6 +20*x^7 +31*x^8 +10*x^9 +5*x^10 +2*x^11 +x^12) / ((1 -x)^4*(1 +x)^4*(1 +x^2)^4). - Colin Barker, Oct 09 2016
Sum_{n>=2} 1/a(n) = 1 - log(2)/2. - Amiram Eldar, Aug 13 2022

More terms from Robert G. Wilson v, Oct 05 2016

A294033 Triangle read by rows, expansion of exp(xz)z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 2, -3, 6, 3, -8, -12, 12, 4, 25, -40, -30, 20, 5, 96, 150, -120, -60, 30, 6, -427, 672, 525, -280, -105, 42, 7, -2176, -3416, 2688, 1400, -560, -168, 56, 8, 12465, -19584, -15372, 8064, 3150, -1008, -252, 72, 9, 79360, 124650, -97920, -51240, 20160, 6300, -1680, -360, 90, 10, -555731, 872960, 685575, -359040, -140910, 44352, 11550, -2640, -495, 110, 11

Offset: 1

Peter Luschny, Oct 24 2017

			Triangle starts:
  [1][   1]
  [2][   2,   2]
  [3][  -3,   6,    3]
  [4][  -8, -12,   12,    4]
  [5][  25, -40,  -30,   20,    5]
  [6][  96, 150, -120,  -60,   30,  6]
  [7][-427, 672,  525, -280, -105, 42, 7]

T(n, 0) = signed A065619. Row sums of abs(T(n,k)) = A231179.
Diagonals A000027, A002378, A027480, A162668.
A003506 (m=1), this seq. (m=2), A294034 (m=3).
Cf. A000111, A247453.

gf := exp(x*z)*z*(tanh(z)+sech(z)):
s := n -> n!*coeff(series(gf,z,n+2),z,n):
C := n -> PolynomialTools:-CoefficientList(s(n),x):
ListTools:-FlattenOnce([seq(C(n), n=1..7)]);
# Alternatively:
T := (n, k) -> `if`(n = k+1, n,
(k+1)*binomial(n,k+1)*2^(n-k-1)*(euler(n-k-1, 1/2)+euler(n-k-1, 1))):
for n from 1 to 7 do seq(T(n,k), k=0..n-1) od;

L[0] := 1; L[n_] := (-1)^Binomial[n, 2] 2 Abs[PolyLog[-n, -I]];
p[n_] := n Sum[Binomial[n - 1, k - 1] L[k - 1] x^(n - k), {k, 0, n}];
Table[CoefficientList[p[n], x], {n, 1, 11}] // Flatten

T(n, k) = (k+1)*binomial(n,k+1)*2^(n-k-1)*(Euler(n-k-1, 1/2) + Euler(n-k-1, 1)) for 0 <= k <= n-2.

T(n, k) is the coefficient of x^k of the polynomial p(n) = n*Sum_{k=1..n} binomial(n-1, k-1)*L(k-1)*x^(n-k) and L(n) = (-1)^binomial(n,2)*A000111(n). In particular n divides T(n, k).

A294034 Triangle read by rows, expansion of exp(xz)z((exp(z) + 1)/((exp(z) + 2exp(-z/2)cos(zsqrt(3)/2))/3) -1), for n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, -4, 12, 12, 4, -15, -20, 30, 20, 5, -54, -90, -60, 60, 30, 6, 133, -378, -315, -140, 105, 42, 7, 792, 1064, -1512, -840, -280, 168, 56, 8, 4293, 7128, 4788, -4536, -1890, -504, 252, 72, 9, -15130, 42930, 35640, 15960, -11340, -3780, -840, 360, 90, 10, -123849, -166430, 236115, 130680, 43890, -24948, -6930, -1320, 495, 110, 11

Offset: 1

Peter Luschny, Oct 24 2017

			Triangle starts:
[1][   1]
[2][   2,    2]
[3][   3,    6,     3]
[4][  -4,   12,    12,    4]
[5][ -15,  -20,    30,   20,    5]
[6][ -54,  -90,   -60,   60,   30,   6]
[7][ 133, -378,  -315, -140,  105,  42,  7]
[8][ 792, 1064, -1512, -840, -280, 168, 56, 8]

Cf. A003506, A294033.

gf := exp(x*z)*z*((exp(z) + 1)/((exp(z) + 2*exp(-z/2)*cos(z*sqrt(3)/2))/3) - 1):
s := n -> n!*coeff(series(gf, z, 12), z, n):
C := n -> PolynomialTools:-CoefficientList(s(n), x):
ListTools:-FlattenOnce([seq(C(n), n=1..11)]);

A344007 Denominators of triangle formed by beginning with 1 on row 1, then producing row n by replacing the largest value on row n-1, k, by 1/n and k - 1/n, and arranging the entries in order from smallest to largest.

Original entry on oeis.org

1, 2, 2, 6, 3, 2, 6, 4, 4, 3, 15, 6, 5, 4, 4, 12, 15, 6, 6, 5, 4, 12, 28, 15, 7, 6, 6, 5, 40, 12, 28, 8, 15, 7, 6, 6, 18, 40, 12, 28, 9, 8, 15, 7, 6, 18, 15, 40, 12, 10, 28, 9, 8, 15, 7, 77, 18, 15, 40, 12, 11, 10, 28, 9, 8, 15, 20, 77, 18, 15, 40, 12, 12, 11, 10, 28, 9, 8

Offset: 1

Evan Lee, Jun 08 2021

If there is more than one copy of the largest entry in row n-1, only one copy is changed.

For a somewhat similar triangle, see Leibniz's Harmonic Triangle A003506. - N. J. A. Sloane, Jun 09 2021

			The triangle's first 10 rows:
  1
  1/2,   1/2
  1/6,   1/3,   1/2
  1/6,   1/4,   1/4,   1/3
  2/15,  1/6,   1/5,   1/4,   1/4
  1/12,  2/15,  1/6,   1/6,   1/5,   1/4
  1/12,  3/28,  2/15,  1/7,   1/6,   1/6,   1/5
  3/40,  1/12,  3/28,  1/8,   2/15,  1/7,   1/6,   1/6
  1/18,  3/40,  1/12,  3/28,  1/9,   1/8,   2/15,  1/7,   1/6
  1/18,  1/15,  3/40,  1/12,  1/10,  3/28,  1/9,   1/8,   2/15,  1/7
  ...
The denominators are:
   1
   2,  2,
   6,  3, 2,
   6,  4, 4, 3,
  15,  6, 5, 4, 4,
  12, 15, 6, 6, 5, 4,
  ...

Cf. A003506.

For numerators see A344008.

lista(nn) = {my(row, nrow, drow); for (n=1, nn, if (n==1, row = [1], k = vecmax(row); nrow = row; nrow[n-1] = 1/n; nrow = concat(nrow, k - 1/n); row = vecsort(nrow);); drow = apply(denominator, row); for (k=1, #drow, print1(drow[k], ", ")););} \\ Michel Marcus, Jun 09 2021

A344008 Triangle of numerators corresponding to A344007.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 2, 1, 4, 1, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 5, 1, 4, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Evan Lee, Jun 09 2021

			The triangle underlying A344007 begins:
  1
  1/2,   1/2
  1/6,   1/3,   1/2
  1/6,   1/4,   1/4,   1/3
  2/15,  1/6,   1/5,   1/4,   1/4
  1/12,  2/15,  1/6,   1/6,   1/5,   1/4
  1/12,  3/28,  2/15,  1/7,   1/6,   1/6,   1/5
  3/40,  1/12,  3/28,  1/8,   2/15,  1/7,   1/6,   1/6
  1/18,  3/40,  1/12,  3/28,  1/9,   1/8,   2/15,  1/7,  1/6
  1/18,  1/15,  3/40,  1/12,  1/10,  3/28,  1/9,   1/8,  2/15,  1/7
  ...
The numerators are:
  1
  1, 1
  1, 1, 1
  1, 1, 1, 1
  2, 1, 1, 1, 1
  1, 2, 1, 1, 1, 1
  1, 3, 2, 1, 1, 1, 1
  3, 1, 3, 1, 2, 1, 1, 1
  1, 3, 1, 3, 1, 1, 2, 1, 1
  1, 1, 3, 1, 1, 3, 1, 1, 2, 1
  ...

Cf. A003506, A344007.

lista(nn) = {my(row, nrow, drow); for (n=1, nn, if (n==1, row = [1], k = vecmax(row); nrow = row; nrow[n-1] = 1/n; nrow = concat(nrow, k - 1/n); row = vecsort(nrow);); drow = apply(numerator, row); for (k=1, #drow, print1(drow[k], ", ")););} \\ Michel Marcus, Jun 09 2021

Corrected by Hugo Pfoertner and Michel Marcus, Jun 09 2021

A046200 Odd numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Original entry on oeis.org

1, 3, 3, 5, 5, 7, 105, 105, 7, 9, 9, 11, 495, 495, 11, 13, 6435, 6435, 13, 15, 1365, 15015, 45045, 45045, 15015, 1365, 15, 17, 17, 19, 2907, 2907, 19, 21, 101745, 101745, 21, 23, 5313, 168245, 1716099, 1716099, 168245, 5313, 23, 25, 18386775, 18386775, 25

Offset: 1

Mohammad K. Azarian

			1/1; --> 1
1/2, 1/2; -->
1/3, 1/6, 1/3; --> 3 3
1/4, 1/12, 1/12, 1/4; --> ...
1/5, 1/20, 1/30, 1/20, 1/5; ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

More terms from James Sellers, Dec 13 1999

A046203 Even numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Original entry on oeis.org

2, 2, 6, 4, 12, 12, 4, 20, 30, 20, 6, 30, 60, 60, 30, 6, 42, 140, 42, 8, 56, 168, 280, 280, 168, 56, 8, 72, 252, 504, 630, 504, 252, 72, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 110, 1320, 2310, 2772, 2310, 1320, 110, 12, 132, 660, 1980, 3960, 5544, 5544

Offset: 1

Mohammad K. Azarian

			1/1; 1/2, 1/2; 1/3, 1/6, 1/3; 1/4, 1/12, 1/12, 1/4; 1/5, 1/20, 1/30, 1/20, 1/5; ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.

Cf. A003506.

More terms from James Sellers, Dec 13 1999

cp's OEIS Frontend

A171631 Triangle read by rows: T(n,k) = n*(binomial(n-2, k-1) + n*binomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

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A229253 Total number of elements of nonempty subsets of divisors of n.

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A237425 Denominators of A164555(n)/A027642(n) + A198631(n)/A006519(n+1).

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A276670 Numerator of (n-1)*n*(n+1)/4.

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A294033 Triangle read by rows, expansion of exp(x*z)*z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

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A294034 Triangle read by rows, expansion of exp(x*z)*z*((exp(z) + 1)/((exp(z) + 2*exp(-z/2)*cos(z*sqrt(3)/2))/3) -1), for n >= 1 and 0 <= k <= n-1.

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A344007 Denominators of triangle formed by beginning with 1 on row 1, then producing row n by replacing the largest value on row n-1, k, by 1/n and k - 1/n, and arranging the entries in order from smallest to largest.

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A344008 Triangle of numerators corresponding to A344007.

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A171631 Triangle read by rows: T(n,k) = n(binomial(n-2, k-1) + nbinomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

A276670 Numerator of (n-1)n(n+1)/4.

A294033 Triangle read by rows, expansion of exp(xz)z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

A294034 Triangle read by rows, expansion of exp(xz)z((exp(z) + 1)/((exp(z) + 2exp(-z/2)cos(zsqrt(3)/2))/3) -1), for n >= 1 and 0 <= k <= n-1.