A004736 -id:A004736 - cp's OEIS frontend

A153583 Convolution triangle by rows, A004736 * (A153582 * 0^(n-k)).

1, 2, 1, 3, 2, 3, 4, 3, 6, 9, 5, 4, 9, 18, 24, 6, 5, 12, 27, 48, 65, 7, 6, 15, 36, 72, 130, 177, 8, 7, 18, 45, 96, 195, 354, 481, 9, 8, 21, 54, 120, 260, 531, 962, 1308, 10, 9, 24, 63, 144, 325, 708, 1443, 2616, 3555, 11, 10, 27, 72, 168, 390, 885, 1924, 3924, 7110, 9664

Offset: 0

Gary W. Adamson, Dec 28 2008

Row sums = A024581: (1, 3, 8, 22, 60, 163,...).

Right border = A153582.

			First few rows of the triangle =
  1;
  2, 1;
  3, 2, 3;
  4, 3, 6, 9;
  5, 4, 9, 18, 24;
  6, 5, 12, 27, 48, 65;
  7, 6, 15, 36, 72, 130, 177;
  8, 7, 18, 45, 96, 195, 354, 481;
  9, 8, 21, 54, 120, 260, 531, 962, 1308;
  10, 9, 24, 63, 144, 325, 708, 1443, 2616, 3555;
  ...
Row 3 = (4, 3, 6, 9) = termwise products of (4, 3, 2, 1) and (1, 1, 3, 9);
where A153582 = (1, 1, 3, 9, 24, 65,...).

Steve Butler, R. L. Graham & Nan Zang, Jumping Sequences, Journal of Integer Sequences, Vol. 11, 2008, 08.4.5.

Cf. A024581, A153582, A004736

tabl(nn) = {my(va = vector(nn), vc = vector(nn)); va[1] = 1; for (n=1, nn, if (n > 1, va[n] = round(exp(1)*va[n-1])); vc[n] = va[n] - sum(k=1, n-1, vc[k]*(n-k+1)); print(vector(n, k, vc[k]*(n-k+1))););} \\ Michel Marcus, Jan 28 2019

Convolution triangle by rows, A004736 * (A153582 * 0^(n-k)).

More terms from Michel Marcus, Jan 28 2019

A154109 Convolution triangle by rows, A004736 * (A154108 * 0^(n-k)); row sums = Bell numbers.

Original entry on oeis.org

1, 2, 0, 3, 0, 2, 4, 0, 4, 7, 5, 0, 6, 14, 27, 6, 0, 8, 21, 54, 114, 7, 0, 10, 28, 81, 228, 523, 8, 0, 12, 35, 108, 342, 1046, 2589, 9, 0, 14, 42, 135, 456, 1569, 5178, 13744, 10, 0, 16, 49, 162, 570, 2092, 7767, 27488, 77821

Offset: 1

Gary W. Adamson, Jan 04 2009

Row sums = Bell numbers, A000110 starting (1, 2, 5, 15, 52, 203, 877,...).

			First few rows of the triangle =
1;
2, 0;
3, 0, 2;
4, 0, 4, 7;
5, 0, 6, 14, 27;
6, 0, 8, 21, 54, 114;
7, 0, 10, 28, 81, 228, 523;
8, 0, 12, 35, 108, 342, 1046, 2589;
9, 0, 14, 42, 135, 456, 1569, 5178, 13744;
10, 0, 16, 49, 162, 570, 2092, 7767, 27488, 77821;
...
Row 5 = (5, 0, 6, 14, 27), sum = A000110(5) = 52 = termwise products of
(5, 4, 3, 2, 1) and (1, 0, 2, 7, 27).

Cf. A154108, A000110.

A004736 * (A154108 * 0^(n-k)); where A004736 = an infinite lower triangular
matrix with (1,2,3,...) in every column and (A154108 * 0^(n-k)) = a matrix
with A154108 (1, 0, 2, 7, 27, 114, 523, 2589...) as the main diagonal
and the rest zeros.

A158841 Triangle read by rows, matrix product of A145677 * A004736.

Original entry on oeis.org

1, 3, 1, 7, 4, 2, 13, 9, 6, 3, 21, 16, 12, 8, 4, 31, 25, 20, 15, 10, 5, 43, 36, 30, 24, 18, 12, 6, 57, 49, 42, 35, 28, 21, 14, 7, 73, 64, 56, 48, 40, 32, 24, 16, 8, 91, 81, 72, 63, 54, 45, 36, 27, 18, 9

Offset: 1

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

			First few rows of the triangle:
    1;
    3,   1;
    7,   4,   2;
   13,   9,   6,   3;
   21,  16,  12,   8,   4;
   31,  25,  20,  15,  10,  5;
   43,  36,  30,  24,  18, 12,  6;
   57,  49,  42,  35,  28, 21, 14,  7;
   73,  64,  56,  48,  40, 32, 24, 16,  8;
   91,  81,  72,  63,  54, 45, 36, 27, 18,  9;
  111, 100,  90,  80,  70, 60, 50, 40, 30, 20, 10;
  133, 121, 110,  99,  88, 77, 66, 55, 44, 33, 22, 11;
  157, 144, 132, 120, 108, 96, 84, 72, 60, 48, 36, 24, 12;
  ...

Cf. A145677, A002061 (column k=1), A158842 (row sums).

A145677 := proc(n,k)
        if n <0 or k < 0 or k > n then
                0;
        elif k = 0 then
                1;
        elif k = n then
                n ;
        else
                0 ;
        end if;
end proc:
A004736 := proc(n,k)
        if n <0 or k < 1 or k > n then
                0;
        else
                n-k+1 ;
        end if;
end proc:
A158841 := proc(n,k)
        add( A145677(n-1,j-1)*A004736(j,k),j=k..n) ;
end proc: # R. J. Mathar, Nov 05 2011

T(n,k) = Sum_{j=k..n} A145677(n-1,j-1)*A004736(j,k), assuming column enumeration k >= 1 in A004736. - R. J. Mathar, Nov 05 2011

A159905 Triangle read by rows, Mobius transform of A004736.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Apr 25 2009

Row sums = A007438, Mobius transform of the triangular numbers.

Left border = phi(n), A000010.

			First few rows of the triangle =
1;
1, 1;
2, 2, 1;
2, 2, 2, 1;
4, 4, 3, 2, 1;
2, 2, 3, 3, 2, 1;
6, 6, 5, 4, 3, 2, 1;
4, 4, 4, 4, 4, 3, 2, 1;
6, 6, 6, 6, 5, 4, 3, 2, 1;
4, 4, 5, 5, 5, 5, 4, 3, 2, 1;
...

Cf. A007438

Triangle read by rows, A054525 * A004736. A054525 = the Mobius transform, A004736 = the natural number "decrescendo" triangle: (1; 2,1; 3,2,1;...)

A171843 Triangle read by rows = truncated columns of an array formed by variants of the natural number decrescendo triangle, A004736.

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 6, 21, 1, 3, 6, 12, 55, 1, 3, 6, 10, 24, 144, 1, 3, 6, 10, 17, 48, 377, 1, 3, 6, 10, 15, 30, 96, 987, 1, 3, 6, 10, 15, 23, 53, 192, 2584, 1, 3, 6, 10, 15, 21, 37, 93, 384, 6765, 1, 3, 6, 10, 15, 21, 30, 61, 163, 768, 17711, 1, 3, 6, 10, 15, 21, 28, 45, 100, 286, 1536, 46368

Offset: 1

Gary W. Adamson, Dec 19 2009

Rows tend to the triangular series, A000217.

Let T(n) be the variants of the natural number decrescendo triangle, A004736; such that T(n) = A004736, prepending n ones to the leftmost column. Then take Lim_{k=1..inf} ((T(n))^k, left-shifted vectors considered as sequences = rows of the array, deleting the first 1. The rows of this triangle sequence are the truncated columns of the array with one "1" per row.

			First few rows of the array are:
.
  1, 3, 8, 21, 55, 144, 377, 987, ...
  1, 1, 3,  6, 12,  24,  48,  96, ...
  1, 1, 1,  3,  6,  10,  17,  30, ...
  1, 1, 1,  1,  3,   6,  10,  15, ...
  1, 1, 1,  1,  1,   3,   6,  10, ...
  ...
First few rows of the triangle =
  1;
  1, 3;
  1, 3, 8;
  1, 3, 6, 21;
  1, 3, 6, 12, 55;
  1, 3, 6, 10, 24, 144;
  1, 3, 6, 10, 17, 48, 377;
  1, 3, 6, 10, 15, 30, 96, 987;
  1, 3, 6, 10, 15, 23, 53, 192, 2584;
  1, 3, 6, 10, 15, 21, 37, 93, 384, 6765;
  1, 3, 6, 10, 15, 21, 30, 61, 163, 768, 17711;
  1, 3, 6, 10, 15, 21, 28, 45, 100, 286, 1536, 46368;
  ...
Example: Row 2 of the array is generated from a variant of A004736, the leftmost column with two prepended 1's, = T(2):
  1;
  1;
  1;
  2, 1;
  3, 2, 1;
  ...
Take lim_{k->inf.} (P(2))^k, obtaining a left-shifted vector considered as a sequence; then delete the first 1, getting row 2 of the array.

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

Row sums are A171844.
Diagonals include A001906, A003945, A259968.
Cf. A004736.

T(n)={[Vec(p) | p<-Vec(sum(k=1, n, x^k*y^(k-1)*(1 - x^k)/((1 - x)*(1 - 2*x + x^2 - x^k)) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Apr 13 2021

a(52) corrected and terms a(56) and beyond from Andrew Howroyd, Apr 13 2021

A193561 Augmentation of the triangle A004736. See Comments.

Original entry on oeis.org

1, 2, 1, 6, 6, 3, 24, 36, 30, 15, 120, 240, 270, 210, 105, 720, 1800, 2520, 2520, 1890, 945, 5040, 15120, 25200, 30240, 28350, 20790, 10395, 40320, 141120, 272160, 378000, 415800, 374220, 270270, 135135, 362880, 1451520, 3175200, 4989600

Offset: 0

Clark Kimberling, Jul 30 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193561, if the triangle is written as (w(n,k)), then
w(n,n)=A001147(n), "double factorial numbers";
w(n,n-1)=A097801(n), (2n)!/(n!*2^(n-1))
col 1:  A000142, n!
col 2: A001286, Lah numbers, (n-1)*n!/2

			First 5 rows of A193560:
1
2.....1
6.....6....3
24....36...30...15
120...240..270..210..105

Cf. A193091.

p[n_, k_] := n + 1 - k
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A004736 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193561 *)
Flatten[Table[v[n], {n, 0, 8}]]

A271863 Recursive sequence based on the central polygonal numbers (A000124) and A004736.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 7, 10, 12, 5, 11, 19, 16, 14, 18, 15, 22, 25, 17, 9, 24, 13, 29, 23, 32, 28, 26, 31, 27, 39, 20, 38, 40, 33, 35, 30, 34, 49, 36, 46, 37, 21, 45, 43, 48, 44, 51, 59, 41, 56, 42, 50, 55, 53, 58, 54, 67, 62, 70, 64, 57, 65, 63, 52, 60, 69, 47

Offset: 1

Max Barrentine, Apr 15 2016

nonn

Conjectured to be a permutation of the natural numbers.

The central polygonal numbers can be constructed by starting with the natural numbers, setting A000124(0)=1 and obtaining A000124(n+1) by reversing the order of the next A000124(n) numbers after A000124(n). This procedure doesn't produce a permutation of the natural numbers for A000124 because the sequence is strictly increasing. The present sequence is constructed by the same procedure, except that a(n+1) is obtained by reversing the next a(A004736(n)) numbers.

			Start with the natural numbers:
1, 2, 3, 4, 5, 6, 7, 8...
a(A004736(1))=1, so reverse the order of the next term, leaving the sequence unchanged:
   (1)
1, (2), 3, 4, 5, 6, 7, 8...
a(A004736(2))=2, so reverse the order of the next 2 terms:
      (2)
1, 2, (4, 3), 5, 6, 7, 8, 9...
a(A004736(3))=1, so reverse the order of the next term, leaving the sequence unchanged:
         (1)
1, 2, 4, (3), 5, 6, 7, 8...
a(A004736(4))=4, so reverse the order of the next 4 terms:
            (4)
1, 2, 4, 3, (8, 7, 6, 5)...
a(A004736(5))=2, so reverse the order of the next 2 terms:
               (2)
1, 2, 4, 3, 8, (6, 7), 5...
a(A004736(6))=1, so reverse the order of the next term, leaving the sequence unchanged:
                  (1)
1, 2, 4, 3, 8, 6, (7), 5...

Max Barrentine, Table of n, a(n) for n = 1..1082

Cf. A000124, A004736.

A127779 Triangle read by rows: A004736 * A127773.

Original entry on oeis.org

1, 2, 3, 3, 6, 6, 4, 9, 12, 10, 5, 12, 18, 20, 15, 6, 15, 24, 30, 30, 21, 7, 18, 30, 40, 45, 42, 28, 8, 21, 36, 50, 60, 63, 56, 36, 9, 24, 42, 60, 75, 84, 84, 72, 45

Offset: 1

Gary W. Adamson, Jan 28 2007

Row sums = bin(n,4), (A000332): (1, 5, 15, 35, ...).
From Clark Kimberling, Sep 16 2008: (Start)
As a rectangular array: R = A000027*A000217; R(m,n) = n*binomial(m+1,2).
R is the accumulation array (cf. A144112) of A002260 (rectangular, with n-th row (n,n,n,n,...)). (End)
As a rectangular array read by ascending antidiagonals, T(n,k) is the total number of triangles obtained when a triangle is cut into n parts with segments going down from the apex to its base and into k parts with segments parallel to its base. See Quora link. - Michel Marcus, Apr 07 2023

			First few rows of the triangle:
  1;
  2,  3;
  3,  6,  6;
  4,  9, 12, 10;
  5, 12, 18, 20, 15;
  6, 15, 24, 30, 30, 21;
  7, 18, 30, 40, 45, 42, 28;
  ...
First few rows of the rectangular array:
  1  3  6 10 15 ...
  2  6 12 20 30 ...
  3  9 18 30 45 ...
  4 12 24 40 60 ...
  5 15 30 50 75 ...
  ...

Quora, How many triangles are in this picture?.

Cf. A004736, A127773, A000217, A000332.

Cf. A002260. - Clark Kimberling, Sep 16 2008

A004736 * A127773 as infinite lower triangular matrices.

A127948 Triangle, A004736 * A127899.

Original entry on oeis.org

1, 0, 2, -1, 1, 3, -2, 0, 2, 4, -3, -1, 1, 3, 5, -4, -2, 0, 2, 4, 6, -5, -3, -1, 1, 3, 5, 7, -6, -4, -2, 0, 2, 4, 6, 8, -7, -5, -3, -1, 1, 3, 5, 7, 9, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10

Offset: 0

Gary W. Adamson, Feb 09 2007

Row sums = n A127899 * A004736 = A002024

			First few rows of the triangle are:
1;
0, 2;
-1, 1, 3;
-2, 0, 2, 4;
-3, -1, 1, 3, 5;
-4, -2, 0, 2, 4, 6;
-5, -3, -1, 1, 3, 5, 7;
...

Cf. A127899, A004736, A002024.

from math import isqrt
def A127948(n): return (m:=n<<1)-(isqrt(m<<2)+1>>1)**2 # Chai Wah Wu, Jun 20 2025

A004736 * A127899 as infinite lower triangular matrices; where A004736 = (1; 2,1; 3,2,1;...).
From Boris Putievskiy, Jan 15 2013: (Start)
a(n) = 2*A000027(n)-A002024(n)^2, n > 0,
a(n) = 2*n-(t+1)^2, where t = floor((-1+sqrt(8*n-7))/2) n > 0. (End)

A128140 A128132 * A004736.

Original entry on oeis.org

1, 3, 2, 7, 5, 3, 13, 10, 7, 4, 21, 17, 13, 9, 5, 31, 26, 21, 16, 11, 6, 43, 37, 31, 25, 19, 13, 7, 57, 50, 43, 36, 29, 22, 15, 8, 73, 65, 57, 49, 41, 33, 25, 17, 9, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10

Offset: 1

Gary W. Adamson, Feb 16 2007

Row sums = A006003, starting (1, 5, 15, 34, 65, 111, ...). Left border = A002061: (1, 3, 7, 13, 21, 31, 43, ...) A128139 = A004736 * A128132

			First few rows of the triangle:
   1;
   3,  2;
   7,  5,  3;
  13, 10,  7,  4;
  21, 17, 13,  9,  5;
  31, 26, 21, 16, 11,  6;
  43, 37, 31, 25, 19, 13,  7;
  ...

Cf. A128132, A004736, A006003, A004006, A128139.

A128132 * A004736 as infinite lower triangular matrices.

cp's OEIS Frontend

A153583 Convolution triangle by rows, A004736 * (A153582 * 0^(n-k)).

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A154109 Convolution triangle by rows, A004736 * (A154108 * 0^(n-k)); row sums = Bell numbers.

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A158841 Triangle read by rows, matrix product of A145677 * A004736.

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A159905 Triangle read by rows, Mobius transform of A004736.

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A171843 Triangle read by rows = truncated columns of an array formed by variants of the natural number decrescendo triangle, A004736.

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PARI

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A193561 Augmentation of the triangle A004736. See Comments.

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Mathematica

A271863 Recursive sequence based on the central polygonal numbers (A000124) and A004736.

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A127779 Triangle read by rows: A004736 * A127773.

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A127948 Triangle, A004736 * A127899.

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