A051162 -id:A051162 - cp's OEIS frontend

A246830 T(n,k) is the concatenation of n-k and n+k in binary; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

0, 3, 2, 10, 7, 4, 15, 20, 13, 6, 36, 29, 22, 15, 8, 45, 38, 31, 40, 25, 10, 54, 47, 72, 57, 42, 27, 12, 63, 104, 89, 74, 59, 44, 29, 14, 136, 121, 106, 91, 76, 61, 46, 31, 16, 153, 138, 123, 108, 93, 78, 63, 80, 49, 18, 170, 155, 140, 125, 110, 95, 144, 113, 82, 51, 20

Offset: 0

Alois P. Heinz, Sep 04 2014

			Triangle T(n,k) begins:
   0
   3  2
  10  7  4
  15 20 13  6
  36 29 22 15  8
  45 38 31 40 25 10
  54 47 72 57 42 27 12
Triangle T(n,k) written in binary (with | denoting the concat operation) begins:
     |0
    1|1      |10
   10|10    1|11     |100
   11|11   10|100   1|101    |110
  100|100  11|101  10|110   1|111    |1000
  101|101 100|110  11|111  10|1000  1|1001  |1010
  110|110 101|111 100|1000 11|1001 10|1010 1|1011 |1100

Alois P. Heinz, Rows n = 0..127, flattened

Column k=0 gives A020330.
T(n+1,n) gives A080565(n+1).
T(2n,n) gives A246831.
Main diagonal gives A005843.
Cf. A007088, A030308, A051162, A025581, A246520 (row maxima).

import Data.Function (on)
a246830 n k = a246830_tabl !! n !! k
a246830_row n = a246830_tabl !! n
a246830_tabl = zipWith (zipWith f) a051162_tabl a025581_tabl where
   f x y = foldr (\b v -> 2 * v + b) 0 $ x |+| y
   (|+|) = (++) `on` a030308_row
-- Reinhard Zumkeller, Sep 04 2014

f:= proc(i, j) local r, h, k; r:=0; h:=0; k:=j;
      while k>0 do r:=r+2^h*irem(k, 2, 'k'); h:=h+1 od; k:=i;
      while k>0 do r:=r+2^h*irem(k, 2, 'k'); h:=h+1 od; r
    end:
T:= (n, k)-> f(n-k, n+k):
seq(seq(T(n, k), k=0..n), n=0..14);

f[i_, j_] := Module[{r, h, k, m}, r=0; h=0; k=j; While[k>0, {k, m} = QuotientRemainder[k, 2]; r = r+2^h*m; h = h+1]; k=i; While[k>0, {k, m} = QuotientRemainder[k, 2]; r = r+2^h*m; h = h+1]; r]; T[n_, k_] := f[n-k, n+k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 03 2016, adapted from Maple *)

A246830 = []
for n in range(10**2):
    for k in range(n):
        A246830.append(int(bin(n-k)[2:]+bin(n+k)[2:],2))
    A246830.append(2*n) # Chai Wah Wu, Sep 05 2014

A070771 b+c+d+e where b>=c>=d>=e>=0 ordered by b then c then d then e.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 3, 4, 5, 4, 5, 6, 6, 7, 8, 3, 4, 5, 6, 5, 6, 7, 7, 8, 9, 6, 7, 8, 8, 9, 10, 9, 10, 11, 12, 4, 5, 6, 7, 6, 7, 8, 8, 9, 10, 7, 8, 9, 9, 10, 11, 10, 11, 12, 13, 8, 9, 10, 10, 11, 12, 11, 12, 13, 14, 12, 13, 14, 15, 16, 5, 6, 7, 8, 7, 8, 9, 9, 10, 11, 8, 9, 10, 10, 11, 12, 11

Offset: 0

Henry Bottomley, May 06 2002

nonn

Cf. A001477, A051162, A070770, A070772 for similar sequences with different numbers of terms summed.

A070772 b+c+d+e+f where b>=c>=d>=e>=f>=0 ordered by b then c then d then e then f.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 4, 5, 6, 7, 6, 7, 8, 8, 9, 10, 3, 4, 5, 6, 7, 5, 6, 7, 8, 7, 8, 9, 9, 10, 11, 6, 7, 8, 9, 8, 9, 10, 10, 11, 12, 9, 10, 11, 11, 12, 13, 12, 13, 14, 15, 4, 5, 6, 7, 8, 6, 7, 8, 9, 8, 9, 10, 10, 11, 12, 7, 8, 9, 10, 9, 10, 11, 11, 12, 13, 10, 11, 12, 12, 13, 14

Offset: 0

Henry Bottomley, May 06 2002

nonn

Cf. A001477, A051162, A070770, A070771 for similar sequences with different numbers of terms summed.

A134478 Triangle read by rows, T(0,0) = 1; n-th row = (n+1) terms of n, n+1, n+2, ...

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 12, 7, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Offset: 0

Gary W. Adamson, Oct 27 2007

Apart from the irregular choice of T(0,0) the same as A051162. - R. J. Mathar, Mar 28 2012

			First few rows of the triangle:
  1;
  1, 2;
  2, 3, 4;
  3, 4, 5,  6;
  4, 5, 6,  7,  8;
  5, 6, 7,  8,  9, 10;
  6, 7, 8,  9, 10, 11, 12;
  7, 8, 9, 10, 11, 12, 13, 14;
  ...

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A051162, A134479 (row sums), A126804 (row products).

Join[{1},Flatten[Table[Range[n,2n],{n,10}]]] (* Harvey P. Dale, Nov 21 2014 *)

concat([1], for(n=1,10, for(k=0,n, print1(n+k, ", ")))) \\ G. C. Greubel, Sep 24 2017

A328901 Triangle T(n, k) read by rows: T(n, k) is the numerator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for n > 0 and T(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 2, 10, 1, 1, 5, 5, 35, 1, 1, 3, 7, 14, 126, 1, 1, 7, 28, 21, 42, 77, 1, 1, 4, 12, 30, 66, 132, 1716, 1, 1, 9, 15, 165, 99, 429, 429, 6435, 1, 1, 5, 55, 55, 143, 1001, 715, 1430, 24310, 1, 1, 11, 22, 143, 1001, 1001, 1144, 2431, 4862, 46189, 1, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 352716

Offset: 0

Stefano Spezia, Oct 30 2019

			n\k|   0   1   2   3   4   5   6
---+----------------------------
0  |   1
1  |   1   1
2  |   1   1   3
3  |   1   1   2  10
4  |   1   1   5   5  35
5  |   1   1   3   7  14 126
6  |   1   1   7  28  21  42  77
...

Stefano Spezia, First 141 rows of the triangle, flattened
D. Armstrong, N. A. Loehr, G. S. Warrington, Rational Parking Functions and Catalan Numbers, Annals of Combinatorics (2016), Volume 20, Issue 1, pp 21-58.
M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.

Main diagonal gives A201058 (for n>0).

Cf. A000108, A046899, A051162, A328902 (denominator).

Flatten[Join[{1},Table[LCM[Binomial[n+k,n],n+k]/(n+k),{n,1,11},{k,0,n}]]]

T(n, k) = lcm(binomial(n + k, n), n + k)/(n + k) for n > 0.

A328902 Triangle T(n, k) read by rows: T(n, k) is the denominator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for 0 <= k <= n, n > 0; T(0, 0) = 1.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Oct 30 2019

			n\k| 0 1 2 3 4 5 6
---+--------------
0  | 1
1  | 1 1
2  | 2 1 2
3  | 3 1 1 3
4  | 4 1 2 1 4
5  | 5 1 1 1 1 5
6  | 6 1 2 3 1 1 1
...

Stefano Spezia, First 141 rows of the triangle, flattened
D. Armstrong, N. A. Loehr, G. S. Warrington, Rational Parking Functions and Catalan Numbers, Annals of Combinatorics (2016), Volume 20, Issue 1, pp 21-58.
M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.

Cf. A000108, A028310 (1st column), A046899, A051162, A328901 (numerator).

Flatten[Join[{1},Table[(n+k)/GCD[n+k,Binomial[n+k,n]],{n,1,12},{k,0,n}]]]

A328902(n,k)=if(n,(n+k)/gcd(binomial(n+k,n),n+k),1) \\ M. F. Hasler, Nov 04 2019

T(n, k) = (n + k)/gcd(binomial(n + k, n), n + k) for n > 0.

A100937 Main diagonal of symmetric square array A100936.

Original entry on oeis.org

1, 3, 14, 76, 435, 2577, 15678, 97272, 612126, 3891890, 24933292, 160663328, 1040074684, 6759228932, 44075916696, 288289595968, 1890894150707, 12434303045721, 81960791460442, 541428229233012, 3583843659376257

Offset: 0

Paul D. Hanna, Nov 23 2004

nonn

			a(3) = 76 = 1*1 + 3^2*2 + 3^2*5 + 1*12 = Sum_{k=0..3} C(3,k)^2*A051163(k).
a(4) = 435 = 1*1 + 4^2*2 + 6^2*5 + 4^2*12 + 1*30 = Sum_{k=0..4} C(4,k)^2*A051163(k).

Cf. A051163, A100936.

PARI

a(n) = Sum_{k=0..n} C(n, k)^2*A051162(k).

A105125 Triangle read by rows: T(n,k) = n^3 + k^3, n >= 0, 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 8, 9, 16, 27, 28, 35, 54, 64, 65, 72, 91, 128, 125, 126, 133, 152, 189, 250, 216, 217, 224, 243, 280, 341, 432, 343, 344, 351, 370, 407, 468, 559, 686, 512, 513, 520, 539, 576, 637, 728, 855, 1024, 729, 730, 737, 756, 793, 854, 945, 1072, 1241, 1458, 1000, 1001, 1008, 1027

Offset: 0

Roger L. Bagula, Apr 09 2005

			Triangle begins (modulo 2 plot is a checkerboard):
  {0}
  {1, 2}
  {8, 9, 16}
  {27, 28, 35, 54}
  {64, 65, 72, 91, 128}
  {125, 126, 133, 152, 189, 250}
  ...
The identity for T(2, 1): 9 = 3*(3^2 + 3*1^2)/4 = 3*12/4 = 9. - _Wolfdieter Lang_, May 15 2015

Cf. A069011. Different from A004999. A257238, A025581, A051162.

seq(seq(n^3+k^3,k=0..n),n=0..10); # Robert Israel, May 15 2015

f[n_, m_, p_] := n^p + m^p p = 3 a = Table[Table[f[n, m, p], {n, 0, m}], {m, 0, 20}] aa = Flatten[a]

T(n,k) = n^3 + k^3, n >= 0, 0 <= k <= n.
T(n, k) = A051162(n, k)*(A051162(n, k)^2 + 3* A025581(n, k)^2)/4. See the comment on A051162 for this identity. - Wolfdieter Lang, May 15 2015
G.f. for triangle: -(9*x^5*y^3 - 8*x^4*y^3 - x^4*y^2 + 7*x^3*y^3 - 36*x^3*y^2 - 2*x^2*y^3 + 5*x^3*y + 27*x^2*y^2 + 12*x^2*y - 8*x*y^2 - x^2 + 3*x*y - 4*x - 2*y - 1)*x/((x-1)^4*(x*y-1)^4). - Robert Israel, May 15 2015

A368045 Triangle read by rows. T(n, k) = (k(k + 1)(2k + 1) + n(n + 1)(2n + 1)) / 6.

Original entry on oeis.org

0, 1, 2, 5, 6, 10, 14, 15, 19, 28, 30, 31, 35, 44, 60, 55, 56, 60, 69, 85, 110, 91, 92, 96, 105, 121, 146, 182, 140, 141, 145, 154, 170, 195, 231, 280, 204, 205, 209, 218, 234, 259, 295, 344, 408, 285, 286, 290, 299, 315, 340, 376, 425, 489, 570

Offset: 0

Peter Luschny, Dec 09 2023

Consider a sequence-to-triangle transformation a -> T, where a is a 0-based sequence and T a regular (0, 0)-based triangular array. The transformation is recursively defined, starting with T(0, 0) = 0, and T(n, n) = a(n) + T(n, n - 1) for n > 0. For k <> n let T(n, k) = a(n) + T(n-1, k).
If a(n) = 1, then T = A051162; if a(n) = n, then T = A367964 (generalizing the triangular numbers); if a(n) = n^2, then T is this triangle.
In the multiplicative form of the transformation, T(0, 0) is set to 1, and the operation '+' is replaced by '*'. For instance, a(n) = 2 is then mapped to T = A368043 and a(n) = n to A143216.

			Triangle T(n, k) starts:
  [0] [  0]
  [1] [  1,   2]
  [2] [  5,   6,  10]
  [3] [ 14,  15,  19,  28]
  [4] [ 30,  31,  35,  44,  60]
  [5] [ 55,  56,  60,  69,  85, 110]
  [6] [ 91,  92,  96, 105, 121, 146, 182]
  [7] [140, 141, 145, 154, 170, 195, 231, 280]
  [8] [204, 205, 209, 218, 234, 259, 295, 344, 408]
  [9] [285, 286, 290, 299, 315, 340, 376, 425, 489, 570]

Cf. A000330 (T(n,0)), A056520 (T(n,1)), A005900 (T(n-1,n)), A006331 (T(n,n)), A094952 (T(2*n,n)), A368046 (row sums), A368047 (alternating row sums).
Cf. A051162 (transform of n^0), A367964 (transform of n^1), this sequence (transform of n^2).
Cf. A368043, A143216.

Module[{n=1},NestList[Append[#+n^2,Last[#]+2(n++^2)]&,{0},10]] (* or *)
Table[(k(k+1)(2k+1)+n(n+1)(2n+1))/6,{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 10 2023 *)

from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [0]
    row = Trow(n - 1) + [0]
    for k in range(n): row[k] += n * n
    row[n] = row[n - 1] + n * n
    return row
print([k for n in range(10) for k in Trow(n)])

T(n, k) = A000330(k) + A000330(n).

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954

G. C. Greubel, Table of n, a(n) for n = 1..350
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly,66 (1959), 375-384.

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023

cp's OEIS Frontend

A246830 T(n,k) is the concatenation of n-k and n+k in binary; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A070771 b+c+d+e where b>=c>=d>=e>=0 ordered by b then c then d then e.

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A070772 b+c+d+e+f where b>=c>=d>=e>=f>=0 ordered by b then c then d then e then f.

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A134478 Triangle read by rows, T(0,0) = 1; n-th row = (n+1) terms of n, n+1, n+2, ...

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A328901 Triangle T(n, k) read by rows: T(n, k) is the numerator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for n > 0 and T(0, 0) = 1.

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A328902 Triangle T(n, k) read by rows: T(n, k) is the denominator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for 0 <= k <= n, n > 0; T(0, 0) = 1.

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A100937 Main diagonal of symmetric square array A100936.

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A105125 Triangle read by rows: T(n,k) = n^3 + k^3, n >= 0, 0 <= k <= n.

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A368045 Triangle read by rows. T(n, k) = (k*(k + 1)*(2*k + 1) + n*(n + 1)*(2*n + 1)) / 6.

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A368045 Triangle read by rows. T(n, k) = (k(k + 1)(2k + 1) + n(n + 1)(2n + 1)) / 6.