A017077 -id:A017077 - cp's OEIS frontend

A213652 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 52, 57, 60, 61, 60, 57, 52, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456, 480, 489, 480, 456

Offset: 0

M. F. Hasler, Jun 17 2012

The n-th row also yields the number of ways to get a total of n, n+1,..., 9n, when summing n integers ranging from 1 to 9.
The row sums equal 9^n = A001019(n).
The row lengths are 1+8n = A017077(n).

			The triangle starts:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1; (row sum = 9, row length = 9)
(row n=2) 1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1; (sum = 81, length = 17)
(row n=3) 1,3,6,10,15,21,28,36,45,52,57,60,61,60,... (sum = 729, length = 25)
(row n=4) 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456,... (sum = 9^4; length = 33),
etc.

Seiichi Manyama, Rows n = 0..49, flattened

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.

Cf. A001019, A017077.

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 9-nomials as a table
r := 9:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do; # Peter Bala, Sep 07 2013

concat(vector(5,k,Vec(sum(j=0,8,x^j)^(k-1))))

T(n,k) = Sum_{i=0..floor(k/9)} (-1)^i*binomial(n,i)*binomial(n+k-1-9*i,n-1) for n >= 0 and 0 <= k <= 8*n. - Peter Bala, Sep 07 2013

A267370 Partial sums of A140091.

Original entry on oeis.org

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510

Offset: 0

Bruno Berselli, Jan 13 2016

After 0, this sequence is the third column of the array in A185874.

Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

			The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

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

Cf. A005581, A051744, A105163, A140091, A185874.

Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).

Magma
```
[n*(n+1)*(n+5)/2: n in [0..50]];
```

Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* Harvey P. Dale, Jul 18 2019 *)

PARI
```
vector(50, n, n--; n*(n+1)*(n+5)/2)
```
Sage
```
[n*(n+1)*(n+5)/2 for n in (0..50)]
```

O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

A277090 Expansion of Product_{k>=0} 1/(1 - x^(8*k+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 30, 31, 34, 38, 41, 43, 44, 45, 45, 46, 50, 55, 60, 63, 65, 66, 67, 68, 72, 79, 85, 90, 93, 95, 96, 98, 103, 111, 120, 127, 132, 135, 137, 139, 145

Offset: 0

Ilya Gutkovskiy, Sep 29 2016

nonn

Number of partitions of n into parts congruent to 1 mod 8.

More generally, the ordinary generating function for the number of partitions of n into parts congruent to 1 mod m (for m>0) is Product_{k>=0} 1/(1 - x^(m*k+1)).

			a(10) = 2, because we have [9, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
Index entries for related partition-counting sequences

Cf. A017077, A284100.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), A109697 (m=5), A109701 (m=6), A109703 (m=7).

CoefficientList[Series[QPochhammer[x, x^8]^(-1), {x, 0, 90}], x]

G.f.: Product_{k>=0} 1/(1 - x^(8*k+1)).
a(n) ~ exp((Pi*sqrt(n))/(2*sqrt(3)))*Gamma(1/8)/(4*3^(1/16)*(2*Pi)^(7/8)*n^(9/16)).
a(n) = (1/n)*Sum_{k=1..n} A284100(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

A004768 Binary expansion ends 001.

Original entry on oeis.org

9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 121, 129, 137, 145, 153, 161, 169, 177, 185, 193, 201, 209, 217, 225, 233, 241, 249, 257, 265, 273, 281, 289, 297, 305, 313, 321, 329, 337, 345, 353, 361, 369, 377, 385, 393, 401, 409, 417, 425, 433, 441, 449, 457, 465, 473, 481, 489

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 28 ).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Complement of A004774.
Cf. A017077.
Cf. A146302. - Reinhard Zumkeller, Oct 30 2008

[8*n + 9: n in [0..60]]; // Vincenzo Librandi, Jul 11 2011

Rest[FromDigits[#,2]&/@(Join[#,{0,0,1}]&/@Tuples[{0,1},7])] (* or *) LinearRecurrence[{2,-1},{9,17},100] (* Harvey P. Dale, May 10 2015 *)

a(n) = 8*n+9 \\ Charles R Greathouse IV, Sep 24 2012

Vec((9 - x) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Jul 04 2019

def a(n): return 8*n + 9
print([a(n) for n in range(61)]) # Michael S. Branicky, Sep 17 2021

From Reinhard Zumkeller, Oct 30 2008: (Start)
a(n) = 8*n + 9.
For n > 0: a(n) = A017077(n-1). (End)
a(n) = 2*a(n-1) - a(n-2); a(0)=9, a(1)=17. - Harvey P. Dale, May 10 2015
G.f.: (9 - x) / (1 - x)^2. - Colin Barker, Jul 04 2019
E.g.f.: exp(x)*(9 + 8*x). - Stefano Spezia, May 13 2021

A371095 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 8*k-7, and A(n+1, k) = R(A(n, k)), n,k >= 1, where Reduced Collatz function R(n) gives the odd part of 3n+1.

Original entry on oeis.org

1, 1, 9, 1, 7, 17, 1, 11, 13, 25, 1, 17, 5, 19, 33, 1, 13, 1, 29, 25, 41, 1, 5, 1, 11, 19, 31, 49, 1, 1, 1, 17, 29, 47, 37, 57, 1, 1, 1, 13, 11, 71, 7, 43, 65, 1, 1, 1, 5, 17, 107, 11, 65, 49, 73, 1, 1, 1, 1, 13, 161, 17, 49, 37, 55, 81, 1, 1, 1, 1, 5, 121, 13, 37, 7, 83, 61, 89, 1, 1, 1, 1, 1, 91, 5, 7, 11, 125, 23, 67, 97

Offset: 1

Antti Karttunen, Apr 24 2024

			Array begins:
n\k|  1   2   3   4   5    6   7   8   9   10  11   12   13   14   15   16
---+------------------------------------------------------------------------
1  |  1,  9, 17, 25, 33,  41, 49, 57, 65,  73, 81,  89,  97, 105, 113, 121,
2  |  1,  7, 13, 19, 25,  31, 37, 43, 49,  55, 61,  67,  73,  79,  85,  91,
3  |  1, 11,  5, 29, 19,  47,  7, 65, 37,  83, 23, 101,  55, 119,   1, 137,
4  |  1, 17,  1, 11, 29,  71, 11, 49,  7, 125, 35,  19,  83, 179,   1, 103,
5  |  1, 13,  1, 17, 11, 107, 17, 37, 11,  47, 53,  29, 125, 269,   1, 155,
6  |  1,  5,  1, 13, 17, 161, 13,  7, 17,  71,  5,  11,  47, 101,   1, 233,
7  |  1,  1,  1,  5, 13, 121,  5, 11, 13, 107,  1,  17,  71,  19,   1, 175,
8  |  1,  1,  1,  1,  5,  91,  1, 17,  5, 161,  1,  13, 107,  29,   1, 263,
9  |  1,  1,  1,  1,  1, 137,  1, 13,  1, 121,  1,   5, 161,  11,   1, 395,
10 |  1,  1,  1,  1,  1, 103,  1,  5,  1,  91,  1,   1, 121,  17,   1, 593,
11 |  1,  1,  1,  1,  1, 155,  1,  1,  1, 137,  1,   1,  91,  13,   1, 445,
12 |  1,  1,  1,  1,  1, 233,  1,  1,  1, 103,  1,   1, 137,   5,   1, 167,
13 |  1,  1,  1,  1,  1, 175,  1,  1,  1, 155,  1,   1, 103,   1,   1, 251,
14 |  1,  1,  1,  1,  1, 263,  1,  1,  1, 233,  1,   1, 155,   1,   1, 377,
15 |  1,  1,  1,  1,  1, 395,  1,  1,  1, 175,  1,   1, 233,   1,   1, 283,
16 |  1,  1,  1,  1,  1, 593,  1,  1,  1, 263,  1,   1, 175,   1,   1, 425,

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A017077 (row 1), A016921 (row 2), A075677.

Cf. also A371096, A371097.

up_to = 91;
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A371095sq(n,k) = if(1==n,8*k-7,R(A371095sq(n-1,k)));
A371095list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A371095sq((a-(col-1)),col))); (v); };
v371095 = A371095list(up_to);
A371095(n) = v371095[n];

A028993 Odd 10-gonal (or decagonal) numbers.

Original entry on oeis.org

1, 27, 85, 175, 297, 451, 637, 855, 1105, 1387, 1701, 2047, 2425, 2835, 3277, 3751, 4257, 4795, 5365, 5967, 6601, 7267, 7965, 8695, 9457, 10251, 11077, 11935, 12825, 13747, 14701, 15687, 16705, 17755, 18837, 19951, 21097, 22275, 23485, 24727, 26001, 27307, 28645

Offset: 0

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Decagonal Number.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001107, A005408, A017077, A028994.

[(2*n+1)*(8*n+1): n in [0..60]]; // Vincenzo Librandi, Oct 18 2013

A028993:=n->(2*n+1)*(8*n+1): seq(A028993(n), n=0..100); # Wesley Ivan Hurt, Apr 26 2017

CoefficientList[Series[-(7 x^2 + 24 x + 1)/(x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Select[PolygonalNumber[10,Range[100]],OddQ] (* or *) LinearRecurrence[{3,-3,1},{1,27,85},50] (* Harvey P. Dale, May 03 2023 *)

a(n)=(2*n+1)*(8*n+1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (2*n+1)*(8*n+1). - N. J. A. Sloane
G.f.: -(7*x^2+24*x+1)/(x-1)^3. - Colin Barker, Nov 18 2012
Sum_{n>=0} 1/a(n) = (4*log(2) + (sqrt(2)+1)*Pi + 2*sqrt(2)*log(1+sqrt(2)))/12. - Amiram Eldar, Feb 27 2022
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: exp(x)*(1 + 26*x + 16*x^2).
a(n) = A005408(n)*A017077(n) = A001107(2*n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139617 a(n) = 136*n + 17.

Original entry on oeis.org

17, 153, 289, 425, 561, 697, 833, 969, 1105, 1241, 1377, 1513, 1649, 1785, 1921, 2057, 2193, 2329, 2465, 2601, 2737, 2873, 3009, 3145, 3281, 3417, 3553, 3689, 3825, 3961, 4097, 4233, 4369, 4505, 4641, 4777, 4913, 5049, 5185, 5321

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 17th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 17th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017077, A057145, A139600.

[17*(8*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

136*Range[0,40]+17  (* or *) LinearRecurrence[{2,-1},{17,153},40] (* Harvey P. Dale, Jul 27 2015 *)

a(n)=136*n+17 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 17*(1+7*x)/(x-1)^2.
E.g.f.: 17*exp(x)*(1 + 8*x).
a(n) = 17*A017077(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A238477 a(n) = 32*n - 27 for n >= 1. Second column of triangle A238475.

Original entry on oeis.org

5, 37, 69, 101, 133, 165, 197, 229, 261, 293, 325, 357, 389, 421, 453, 485, 517, 549, 581, 613, 645, 677, 709, 741, 773, 805, 837, 869, 901, 933, 965, 997, 1029, 1061, 1093, 1125, 1157, 1189, 1221, 1253, 1285, 1317, 1349, 1381, 1413, 1445, 1477, 1509, 1541, 1573

Offset: 1

Wolfdieter Lang, Mar 10 2014

This sequence gives all start numbers a(n) (sorted increasingly) of Collatz sequences of length 6 following the pattern udddd = ud^4, with u (for 'up'), mapping an odd number m to 3*m+1, and d (for 'down'), mapping an even number m to m/2. The last entry of this sequence is required to be odd and it is given by 6*n - 5.

This appears in Example 2.1. for x = 4 in the M. Trümper paper given as a link below.

			a(1) = 5 because the Collatz sequence of length 6 is [5, 16, 8, 4, 2, 1], following the pattern udddd, ending in 1, and 5 is the smallest start number following this pattern ending in an odd number.
a(2) = 37 with the length 6 Collatz sequence [37, 112, 56, 28, 14, 7] ending in 12 - 5 = 7, and this is the second smallest start number with this sequence pattern ending in an odd number.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017077 (first column), A238475, A239123 (third column).

CoefficientList[Series[(5 + 27 x)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)

O.g.f.: x*(5+27*x)/(1-x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 27 + exp(x)*(32*x - 27).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A326296 Triangle of numbers T(n,k) = 2floor(k/2)(n-k) + ceiling((k-1)^2/2), 1<=k<=n.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 0, 5, 4, 5, 0, 7, 6, 9, 8, 0, 9, 8, 13, 12, 13, 0, 11, 10, 17, 16, 19, 18, 0, 13, 12, 21, 20, 25, 24, 25, 0, 15, 14, 25, 24, 31, 30, 33, 32, 0, 17, 16, 29, 28, 37, 36, 41, 40, 41, 0, 19, 18, 33, 32, 43, 42, 49, 48, 51, 50, 0, 21, 20, 37, 36, 49, 48, 57, 56, 61, 60, 61

Offset: 1

M. Ryan Julian Jr., Sep 10 2019

T(n,k) gives the maximum number of inversions in a permutation on n symbols containing a single k-cycle and (n-k) other fixed points.
T(n,n) = A000982(n).
T(n,n-1) = A097063(n).

			Triangle begins:
0;
0, 1;
0, 3, 2;
0, 5, 4, 5;
0, 7, 6, 9, 8;
0, 9, 8, 13, 12, 13;
0, 11, 10, 17, 16, 19, 18;
0, 13, 12, 21, 20, 25, 24, 25;
0, 15, 14, 25, 24, 31, 30, 33, 32;
0, 17, 16, 29, 28, 37, 36, 41, 40, 41;
0, 19, 18, 33, 32, 43, 42, 49, 48, 51, 50;
0, 21, 20, 37, 36, 49, 48, 57, 56, 61, 60, 61;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Diagonals give A000982, A097063, A326657, A326658.
Columns give A000004, A005408, A005843, A016813, A008586, A016921, A008588, A017077, A008590.
Row sums give A000330.

T(n,k) = {2*floor(k/2)*(n-k) + ceil((k-1)^2/2)} \\ Andrew Howroyd, Sep 10 2019

T(n,k) = 2*floor(k/2)*(n-k) + ceiling((k-1)^2/2).

T(n,k) = 2*floor(k/2)*(n-k) + binomial(k,2) - ceiling(k/2) + 1.

cp's OEIS Frontend

A213652 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...

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A267370 Partial sums of A140091.

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A277090 Expansion of Product_{k>=0} 1/(1 - x^(8*k+1)).

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A004768 Binary expansion ends 001.

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A371095 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 8*k-7, and A(n+1, k) = R(A(n, k)), n,k >= 1, where Reduced Collatz function R(n) gives the odd part of 3n+1.

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A028993 Odd 10-gonal (or decagonal) numbers.

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A139617 a(n) = 136*n + 17.

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A326296 Triangle of numbers T(n,k) = 2floor(k/2)(n-k) + ceiling((k-1)^2/2), 1<=k<=n.