A099375 -id:A099375 - cp's OEIS frontend

A143865 Eigentriangle of A099375 (odd number subsequences decrescendo).

1, 3, 1, 5, 3, 4, 7, 5, 12, 12, 9, 7, 20, 36, 36, 11, 9, 28, 60, 108, 108, 13, 11, 36, 84, 180, 324, 324, 15, 13, 44, 108, 252, 540, 972, 972, 17, 15, 52, 132, 324, 756, 1620, 2916, 2916

Offset: 1

Gary W. Adamson, Sep 04 2008

The product A099375 * (A003946 * 0^(n-k)) creates a new eigentriangle with rows = termwise product of odd descrendo terms, e.g.: (7, 5, 3, 1) and first n terms of (1, 1, 4, 12,). The resulting triangle has row sums = (1, 4, 12, 36, 108,...), right border = (1, 1, 4, 12, 36,...) and the property that sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle = 1; 3, 1; 5, 3, 4; 7, 5, 12, 12; 9, 7, 20, 36, 36; 11, 9, 28, 60, 108, 108; 13, 11, 36, 84, 180, 324, 324; ... Row 4 = termwise product of (7, 5, 3, 1) and (1, 1, 4, 12) = (7*1, 5*1, 3*4, 1*12).

A099375, Cf. A003946

Triangle read by rows, A099375: (1; 3,1; 5,3,1;...) * (A003946 * 0^(n-k)); 1<=k<=n. (A003946 * 0^(n-k) = an infinite lower triangular matrix with shifted A003946: (1, 1, 4, 12, 36, 108,...) in the main diagonal and the rest zeros.

A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, 1, 3, 5, 7, 9, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23

Offset: 1

Paul Curtz, Mar 18 2009

Row sums are n^2 = A000290(n).
The triangle sums, see A180662 for their definitions, link this triangle of odd numbers with seventeen different sequences, see the crossrefs. The knight sums Kn14 - Kn110 have been added. - Johannes W. Meijer, Sep 22 2010
A208057 is the eigentriangle of A158405 such that as infinite lower triangular matrices, A158405 * A208057 shifts the latter, deleting the right border of 1's. - Gary W. Adamson, Feb 22 2012
T(n,k) = A099375(n-1,n-k), 1<=k<=n. [Reinhard Zumkeller, Mar 31 2012]

			The triangle contains the first n odd numbers in row n:
  1;
  1,3;
  1,3,5;
  1,3,5,7;
From _Seiichi Manyama_, Dec 02 2017: (Start)
    |       a(n)        |                               | A000290(n)
   -----------------------------------------------------------------
   0|                                                      (=  0)
   1|                 1 = 1/3 * ( 3)                       (=  1)
   2|             1 + 3 = 1/3 * ( 5 +  7)                  (=  4)
   3|         1 + 3 + 5 = 1/3 * ( 7 +  9 + 11)             (=  9)
   4|     1 + 3 + 5 + 7 = 1/3 * ( 9 + 11 + 13 + 15)        (= 16)
   5| 1 + 3 + 5 + 7 + 9 = 1/3 * (11 + 13 + 15 + 17 + 19)   (= 25)
(End)

Seiichi Manyama, Rows n = 1..140, flattened
Daniel Erman, The Josephus Problem, Numberphile video (2016)
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Index entries for sequences related to the Josephus Problem

Cf. A129326, A099375, A005408.
Triangle sums (see the comments): A000290 (Row1; Kn11 & Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A000027 (Row2); A005563 (Kn12); A028347 (Kn13); A028560 (Kn14); A028566 (Kn15); A098603 (Kn16); A098847 (Kn17); A098848 (Kn18); A098849 (Kn19); A098850 (Kn110); A000217 (Kn21. Kn22, Kn23, Fi2, Ze2); A000384 (Kn3, Fi1, Ze3); A000212 (Ca2 & Ze4); A000567 (Ca3, Ze1); A011848 (Gi2); A001107 (Gi3). - Johannes W. Meijer, Sep 22 2010
Cf. A063656, A063657, A208057, A292610, A292611.

a158405 n k = a158405_row n !! (k-1)
a158405_row n = a158405_tabl !! (n-1)
a158405_tabl = map reverse a099375_tabl
-- Reinhard Zumkeller, Mar 31 2012

Table[2 Range[1, n] - 1, {n, 12}] // Flatten (* Michael De Vlieger, Oct 01 2015 *)

a(n) = 2*(n-floor((-1+sqrt(8*n-7))/2)*(floor((-1+sqrt(8*n-7))/2)+1)/2)-1;
vector(100, n, a(n)) \\ Altug Alkan, Oct 01 2015

a(n) = 2*i-1, where i = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 03 2013

a(n) = 2*A002262(n-1) + 1. - Eric Werley, Sep 30 2015

Edited by R. J. Mathar, Oct 06 2009

A152204 Triangle read by rows: T(n,k) = 2n-4k+5 (n >= 0, 1 <= k <= 1+floor(n/2)).

Original entry on oeis.org

1, 3, 5, 1, 7, 3, 9, 5, 1, 11, 7, 3, 13, 9, 5, 1, 15, 11, 7, 3, 17, 13, 9, 5, 1, 19, 15, 11, 7, 3, 21, 17, 13, 9, 5, 1, 23, 19, 15, 11, 7, 3, 25, 21, 17, 13, 9, 5, 1, 27, 23, 19, 15, 11, 7, 3, 29, 25, 21, 17, 13, 9, 5, 1, 31, 27, 23, 19, 15, 11, 7, 3, 33

Offset: 0

Gary W. Adamson, Nov 29 2008

All terms are odd, decreasing across rows. Row sums = A000217, the triangular numbers.
From Johannes W. Meijer, Sep 08 2013: (Start)
Triangle read by rows formed from the antidiagonals of triangle A099375.
The alternating row sums equal A098181(n). (End)

			First few rows of the triangle:
  1
  3
  5  1
  7  3
  9  5  1
  11 7  3
  13 9  5  1
  15 11 7  3
  17 13 9  5 1
  19 15 11 7 3
  21 17 13 9 5 1
  ...

Nathaniel Johnston, Rows n = 0..200 of irregular triangle, flattened

Cf. A000217.

T := proc(n,k) return 2*n-4*k+5: end: seq(seq(T(n,k), k=1..1+floor(n/2)), n=0..20); # Nathaniel Johnston, May 01 2011

By columns, odd terms in every column, n-th column starts at row (2*n).
From Johannes W. Meijer, Sep 08 2013: (Start)
T(n, k) = A099375(n-k+1, k-1), n >= 0 and 1 <= k <= 1+floor(n/2).
T(n, k) = A158405(n+1, n-2*k+2). (End)

Edited by N. J. A. Sloane, Sep 25 2010, following a suggestion from Emeric Deutsch

Offset corrected by Johannes W. Meijer, Sep 07 2013

A208904 Triangle of coefficients of polynomials v(n,x) jointly generated with A208660; see the Formula section.

Original entry on oeis.org

1, 3, 1, 5, 6, 1, 7, 19, 9, 1, 9, 44, 42, 12, 1, 11, 85, 138, 74, 15, 1, 13, 146, 363, 316, 115, 18, 1, 15, 231, 819, 1059, 605, 165, 21, 1, 17, 344, 1652, 2984, 2470, 1032, 224, 24, 1, 19, 489, 3060, 7380, 8378, 4974, 1624, 292, 27, 1, 21, 670, 5301, 16488

Offset: 1

Clark Kimberling, Mar 03 2012

For a discussion and guide to related arrays, see A208510.
Riordan array ((1+x)/(1-x)^2, x(1+x)/(1-x)^2) (follows from Kruchinin formula). - Ralf Stephan, Jan 02 2014
From Peter Bala, Jul 21 2014: (Start)
Let M denote the lower unit triangular array A099375 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

			First five rows:
1
3...1
5...6....1
7...19...9....1
9...44...42...12...1
First five polynomials v(n,x):
1
3 + x
5 + 6x + x^2
7 + 19x + 9x^2 + x^3
9 + 44x + 42x^2 + 12x^3 + x^4
From _Peter Bala_, Jul 21 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
/1        \/1        \/1        \      /1            \
|3 1      ||0 1      ||0 1      |      |3  1         |
|5 3 1    ||0 3 1    ||0 0 1    |... = |5  6  1      |
|7 5 3 1  ||0 5 3 1  ||0 0 3 1  |      |7 19  9  1   |
|9 7 5 3 1||0 7 5 3 1||0 0 5 3 1|      |9 44 42 12 1 |
|...      ||...      ||...      |      |...
(End)

Cf. A208660, A208510. A099375.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208660 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208904 *)

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
From Vladimir Kruchinin, Mar 11 2013: (Start)
T(n,k) = sum(i=0..n, binomial(i+k-1,2*k-1)*binomial(k,n-i))
((x+x^2)/(1-x)^2)^k = sum(n>=k, T(n,k)*x^n).
T(n,2)=A005900(n).
T(2*n-1,n) / n = A003169(n).
T(2*n,n) = A156894(n), n>1.
sum(k=1..n, T(n,k)) = A003946(n).
sum(k=1..n, T(n,k)*(-1)^(n+k)) = A078050(n).
n*sum(k=1..n, T(n,k)/k) = A058481(n). (End)
Recurrence: T(n+1,k+1) = sum {i = 0..n-k} (2*i + 1)*T(n-i,k). - Peter Bala, Jul 21 2014

A261693 Irregular triangle read by rows in which row n lists the positive odd numbers in decreasing order starting with 2^n - 1. T(0, 1) = 0 and T(n, k) for n >= 1, 1 <= k <= 2^(n-1).

Original entry on oeis.org

0, 1, 3, 1, 7, 5, 3, 1, 15, 13, 11, 9, 7, 5, 3, 1, 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 63, 61, 59, 57, 55, 53, 51, 49, 47, 45, 43, 41, 39, 37, 35, 33, 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 127, 125, 123, 121, 119, 117, 115, 113, 111, 109, 107, 105, 103, 101, 99, 97, 95, 93

Offset: 0

Omar E. Pol, Sep 25 2015

Also the first differences of A261692.
Number of cells turned ON at n-th stage of the cellular automaton of A261692.
This irregular triangle A (instead of T) appears also in the linearization of the following product of Chebyshev T polynomials (A053120): PrT(n) := Product_{j=1..n} T(2^j, x)  = (1/2^(n-1))*Sum_{k=1..2^(n-1)} T(2*A(n, k), x), for n >= 1. Proof via 2*T(n, x)*T(m, x) = T(n+m, x) + T(|n-m|, x). - Wolfdieter Lang, Oct 26 2019

			With the terms written as an irregular triangle T in which row lengths are the terms of A011782 the sequence begins:
0;
1;
3, 1;
7, 5, 3, 1;
15, 13, 11, 9, 7, 5, 3, 1;
31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1;
...
-------------------------------------------------------------------------------
From _Wolfdieter Lang_, Oct 26 2019: (Start)
Chebyshev T(2^j)-products (the argument x is here omitted):
n = 1: T(2) = (2^0)*T(2*1),
n = 2: T(2)*T(4) = (1/2)*(T(2*3) + T(2*1)) = (T(6) + T(2))/2,
n = 3: T(2)*T(4)*T(8) =  (1/2^2)*(T(2*7) + T(2*5) + T(2*3) + T(2*1))
       = (T(14) + T(10) + T(6) + T(2))/4.
... (End)

Cf. A036563, A005408, A011782, A080079, A099375, A147582, A262617, A261692, A262621.

Column 1 is A000225. Row sums give A000302, n >= 1.

A261693 := n -> Bits:-Nor(2*n, 2*n):
seq(A261693(n), n=0..81); # Peter Luschny, Sep 23 2019

Table[Reverse[2 Range[2^(n - 1)] - 1], {n, 0, 7}] /. {} -> 0 // Flatten (* Michael De Vlieger, Oct 05 2015 *)

tabf(nn) = {for (n=0, nn, print1(n, ":"); for (k=1, 2^(n-2), print1(2^(n-1) - 2*k + 1, ", ");); print(););} \\ Michel Marcus, Oct 27 2015

T(n, k) = 2^n + 1 - 2*k, n >= 1, 1 <= k <= 2^(n-1), and T(0, 0) = 0.

As a sequence: a(n) = A262621(n)/4, n >= 1, and a(0) = 0.

Corrections by Wolfdieter Lang, Nov 15 2019

A128225 A127899 (unsigned) * A004736.

Original entry on oeis.org

1, 6, 2, 15, 9, 3, 28, 20, 12, 4, 45, 35, 25, 15, 5, 66, 54, 42, 30, 18, 6, 91, 77, 63, 49, 35, 21, 7, 120, 104, 88, 72, 56, 40, 24, 8, 153, 135, 117, 99, 81, 63, 45, 27, 9, 190, 170, 150, 130, 110, 90, 70, 50, 30, 10

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = the cubes, A000578: (1, 8, 27, 64, 125, ...). Left column = the hexagonal numbers: A000384: (1, 6, 15, 28, ...). A128226 = A004736 * A127899.

			First few rows of the triangle are:
   1;
   6,  2;
  15,  9,  3;
  28, 20, 12,  4;
  45, 35, 25, 15,  5;
  66, 54, 42, 30, 18,  6;
  91, 77, 63, 49, 35, 21,  7;
  ...

Cf. A000384, A000578, A004736, A099375, A127899, A128226.

(* a127899U computes the unsigned version of A127899 *)
a127899U[n_, k_] := If[n==k||n-1==k, n, 0]/;(1<=k<=n)
a004736[n_, k_] := n-k+1/;(1<=k<=n+1)
a128225[n_, k_] := a127899U[n, n](a004736[n, k] + a004736[n-1, k])/;(1<=k<=n)
a128225[r_] := Table[a128225[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128225[7]] (* triangle *)
Flatten[a128225[10]] (* data *) (* Hartmut F. W. Hoft, Mar 13 2017 *)

A127899 (unsigned) * A004736, as infinite lower triangular matrices. Triangle read by rows: n*[(1); (3,1); (5,3,1);...]; cf. A099375.

A101038 Inverse to sequence matrix for odd numbers.

Original entry on oeis.org

1, -3, 1, 4, -3, 1, -4, 4, -3, 1, 4, -4, 4, -3, 1, -4, 4, -4, 4, -3, 1, 4, -4, 4, -4, 4, -3, 1, -4, 4, -4, 4, -4, 4, -3, 1, 4, -4, 4, -4, 4, -4, 4, -3, 1, -4, 4, -4, 4, -4, 4, -4, 4, -3, 1, 4, -4, 4, -4, 4, -4, 4, -4, 4, -3, 1, -4, 4, -4, 4, -4, 4, -4, 4, -4, 4, -3, 1, 4, -4, 4, -4

Offset: 0

Paul Barry, Jan 22 2005

Riordan matrix ((1-x)^2/(1+x), x). Inverse matrix is A099375 Row sums yield (-1)^n*A040000. Diagonal sums are (-1)(2n+1)=(-1)^n*A005408.

			Rows begin {1}, {-3,1}, {4,-3,1}, {-4,4,-3,1}, {4,-4,4,-3,1},...

Number triangle T(n, k)=if(k<=n, 4(-1)^(n-k)-3*0^(n-k)+C(1, n-k)-C(0, n-k), 0)

A104734 Triangle T(n,k) = sum_{j=k..n} (2n-2j+1)*binomial(k,j-k), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 3, 1, 5, 4, 1, 7, 8, 5, 1, 9, 12, 12, 6, 1, 11, 16, 20, 17, 7, 1, 13, 20, 28, 32, 23, 8, 1, 15, 24, 36, 48, 49, 30, 9, 1, 17, 28, 44, 64, 80, 72, 38, 10, 1, 19, 32, 52, 80, 112, 129, 102, 47, 11, 1, 21, 36, 60, 96, 144, 192, 201, 140, 57, 12, 1, 23, 40, 68, 112, 176, 256, 321, 303, 187, 68, 13, 1, 25, 44, 76, 128, 208, 320, 448, 522, 443, 244, 80, 14, 1

Offset: 0

Gary W. Adamson, Mar 20 2005

Array A210489 (without first row) read downwards antidiagonals. - R. J. Mathar, Sep 17 2013

			First few rows of the triangle are:
1;
3, 1;
5, 4, 1;
7, 8, 5, 1;
9, 12, 12, 6, 1;
11, 16, 20, 17, 7, 1;
...

Cf. A001891 (row sums), A026729.

Matrix product of the triangle A = A099375 by B = [1; 0, 1; 0, 1, 1; 0, 0, 2, 1; 0, 0, 1, 3, 1;...] (which is the triangular view of A026729).

A152203 Triangle T(n,k) = (2n+1-2k)*fibonacci(k), read by rows.

Original entry on oeis.org

1, 3, 1, 5, 3, 2, 7, 5, 6, 3, 9, 7, 10, 9, 5, 11, 9, 14, 15, 15, 8, 13, 11, 18, 21, 25, 24, 13, 15, 13, 22, 27, 35, 40, 39, 21, 17, 15, 26, 33, 45, 56, 65, 63, 34, 19, 17, 30, 39, 55, 72, 91, 105, 102, 55, 21, 19, 34, 45, 65, 88, 117, 147, 170, 165

Offset: 1

Gary W. Adamson, Nov 29 2008

			First few rows of the triangle =
1;
3, 1;
5, 3, 2;
7, 5, 6, 3;
9, 7, 10, 9, 5;
11, 9, 14, 15, 15, 8;
13, 11, 18, 21, 25, 24, 13;
15, 13, 22, 27, 35, 40, 39, 21;
17, 15, 26, 33, 45, 56, 65, 63, 34;
...
Row 4 = (7, 5, 6, 3) = termwise products of (7, 5, 3, 1) and (1, 1, 2, 3).

Cf. A099375, A127647, A001891 (row sums).

Flatten[Table[(2n+1-2k)Fibonacci[k],{n,15},{k,n}]] (* Harvey P. Dale, Mar 15 2015 *)

Triangle read by rows, A099375 * A127647 = termwise products of odd numbers and the Fibonacci series.

cp's OEIS Frontend

A143865 Eigentriangle of A099375 (odd number subsequences decrescendo).

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A158405 Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m

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A152204 Triangle read by rows: T(n,k) = 2*n-4*k+5 (n >= 0, 1 <= k <= 1+floor(n/2)).

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A208904 Triangle of coefficients of polynomials v(n,x) jointly generated with A208660; see the Formula section.

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A261693 Irregular triangle read by rows in which row n lists the positive odd numbers in decreasing order starting with 2^n - 1. T(0, 1) = 0 and T(n, k) for n >= 1, 1 <= k <= 2^(n-1).

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A128225 A127899 (unsigned) * A004736.

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A101038 Inverse to sequence matrix for odd numbers.

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A104734 Triangle T(n,k) = sum_{j=k..n} (2n-2j+1)*binomial(k,j-k), read by rows, 0<=k<=n.

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A152204 Triangle read by rows: T(n,k) = 2n-4k+5 (n >= 0, 1 <= k <= 1+floor(n/2)).