A014631 -id:A014631 - cp's OEIS frontend

A265912 Smallest m such that A014631(n) occurs in row m of Pascal's triangle.

0, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17

Offset: 1

Reinhard Zumkeller, Dec 18 2015

nonn

Each n occurs A126257(n) times consecutively.

			First occurrences of z(n)=A014631(n) in the left part of Pascal's triangle, repetitions marked:
.   0: z(1)                                       [1]
.   1: *z(1)                                      [1]
.   2: *z(1)  z(2)                                [1,2]
.   3: *z(1)  z(3)                                [1,3]
.   4: *z(1)  z(4)  z(5)                          [1,4,6]
.   5: *z(1)  z(6)  z(7)                          [1,5,10]
.   6: *z(1) *z(5)  z(8)  z(9)                    [1,6,15,20]
.   7: *z(1)  z(10) z(11) z(12)                   [1,7,21,35]
.   8: *z(1)  z(13) z(14) z(15) z(16)             [1,8,28,56,70]
.   9: *z(1)  z(17) z(18) z(19) z(20)             [1,9,36,84,126]
.  10: *z(1) *z(7)  z(21) z(22) z(23) z(24)       [1,10,45,120,210,252]
.  11: *z(1)  z(25) z(26) z(27) z(28) z(29)       [1,11,55,165,330,462]
.  12: *z(1)  z(30) z(31) z(32) z(33) z(34) z(35) [1,12,66,220,495,792,924]
---------------------------------------------------------------------------
.    n: 1  2  3  4  5  6   7   8   9  10  11  12  13  14  15  16  17  18
. z(n): 1  2  3  4  6  5  10  15  20   7  21  35   8  28  56  70   9  36

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007318, A014631, A119629.

import Data.List (findIndex); import Data.Maybe (fromJust)
a265912 = fromJust . (flip findIndex a007318_tabl) . elem . a014631

from itertools import count, islice
def A265912_gen(): # generator of terms
    s, c =(1,), set()
    for i in count(0):
        for d in s:
            if d not in c:
                yield i
                c.add(d)
        s=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + ((s[-1]<<1,) if i&1 else ())
A265912_list = list(islice(A265912_gen(),30)) # Chai Wah Wu, Oct 17 2023

A119629 Inverse permutation to sequence A014631.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 13, 17, 7, 25, 30, 36, 42, 8, 54, 61, 69, 78, 9, 11, 104, 115, 126, 138, 150, 163, 14, 189, 203, 218, 233, 249, 265, 12, 18, 315, 333, 352, 371, 391, 411, 432, 453, 21, 496, 519, 542, 566, 590, 615, 640, 666, 692, 26, 15, 771, 799, 828, 857, 887

Offset: 1

Leroy Quet, Jun 08 2006

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A014631.

Cf. A265912.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a119629 = (+ 1) . fromJust . (`elemIndex` a014631_list)
-- Reinhard Zumkeller, Dec 18 2015

lst = {1}; t = Flatten[Table[Binomial[n, m], {n, 16}, {m, Floor[n/2]}]]; Do[ If[ !MemberQ[lst, t[[n]]], AppendTo[lst, t[[n]] ]], {n, Length@t}]; Flatten@Table[ Position[lst, n], {n, 61}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jun 08 2006

A119741 A008279, with the first and last of each row removed.

Original entry on oeis.org

2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 6, 30, 120, 360, 720, 7, 42, 210, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800

Offset: 2

Lekraj Beedassy, Jul 29 2006

Triangle read by rows: T(n,k) (n>=2, k=1..n-1) is the number of topologies t on n points having exactly k+2 open sets such that t contains exactly one open set of size m for each m in {0,1,2,...,s,n} where s is the size of the largest proper open set in t. - N. J. A. Sloane, Jan 29 2016 [clarified by Geoffrey Critzer, Feb 19 2017]

			Triangle begins:
   2;
   3,  6;
   4, 12,  24;
   5, 20,  60,  120;
   6, 30, 120,  360,   720;
   7, 42, 210,  840,  2520,   5040;
   8, 56, 336, 1680,  6720,  20160,  40320;
   9, 72, 504, 3024, 15120,  60480, 181440,  362880;
  10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Row sums give A038156.
Triangles in this series: A268216, A268217, A268221, A268222, A268223.
Cf. A008279, A014631, A282507.

T:= (n, k)-> n!/(n-k)!:
seq(seq(T(n,k), k=1..n-1), n=2..11);  # Alois P. Heinz, Aug 22 2025

Table[FactorialPower[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 21 2020 *)

a(n) = (A003057(n))!/(A004736(n))! = (A002260(n))!*(A014410(n)).

T(n,k) = A173333(n+1,n-k+1), 1<=k<=n. - Reinhard Zumkeller, Feb 19 2010

Edited by Don Reble, Aug 01 2006

A132311 Triangle read by rows: T(n,k) is the number of partitions of binomial(n,k) into parts of the first n rows of Pascal's triangle, 0<=k<=n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 7, 4, 1, 1, 6, 28, 28, 6, 1, 1, 11, 117, 318, 117, 11, 1, 1, 14, 388, 3344, 3344, 388, 14, 1, 1, 21, 1757, 71277, 290521, 71277, 1757, 21, 1, 1, 29, 8270, 2031198, 53679222, 53679222, 2031198, 8270, 29, 1, 1, 42, 40243, 74464383, 19465193506, 147286801214, 19465193506, 74464383, 40243, 42, 1

Offset: 0

Reinhard Zumkeller, Aug 18 2007

T(n,k) = T(n,n-k).
T(n,0) = 1 for n>0.
A000041(n) - 1 <= T(n,1) <= A000041(n) for n>1.

			A007318(4,2) = A007318(6,1) = 6: T(4,2) = #{3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1} = 7, but T(6,1) = A000041(6) = 11.
Triangle T(n,k) begins:
  0;
  1,  1;
  1,  1,    1;
  1,  2,    2,     1;
  1,  4,    7,     4,      1;
  1,  6,   28,    28,      6,     1;
  1, 11,  117,   318,    117,    11,    1;
  1, 14,  388,  3344,   3344,   388,   14,  1;
  1, 21, 1757, 71277, 290521, 71277, 1757, 21, 1;
  ...

Alois P. Heinz, Rows n = 0..18, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A132312, A007318, A126257, A014631.

A132312 Triangle read by rows: T(n,k) = number of partitions of binomial(n,k) into distinct parts of the first n rows of Pascal's triangle, 0<=k<=n.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 4, 7, 6, 7, 4, 1, 1, 4, 11, 14, 14, 11, 4, 1, 1, 5, 28, 57, 56, 57, 28, 5, 1, 1, 7, 73, 273, 434, 434, 273, 73, 7, 1, 1, 10, 189, 1411, 3479, 3980, 3479, 1411, 189, 10, 1, 1, 11, 300, 4138, 16293, 26555, 26555, 16293, 4138, 300, 11, 1

Offset: 0

Reinhard Zumkeller, Aug 18 2007

T(n,k) = T(n,n-k);
T(n,0) = 1 for n>0;
A000009(n) - 1 <= T(n,1) <= A000009(n) for n>1;

			T(9,1) = A000009(9)-1 = 7;
A007318(5,2) = A007318(10,1) = 10:
T(5,2) = #{6+4, 6+3+1, 4+3+2+1} = 3,
but T(10,1) = A000009(10) = 10.

Alois P. Heinz, Rows n = 0..28, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A132311, A007318, A126257, A014631.

T[n_] := T[n] = Table[Binomial[m, k], {m, 0, n-1}, {k, 0, m}] // Flatten // Union;
T[n_, k_] /; k <= n/2 := T[n, k] = Select[ IntegerPartitions[ Binomial[n, k], Length[T[n]], T[n]], Length[#] == Length[Union[#]]&] // Length;
T[n_, k_] /; k > n/2 := T[n, k] = T[n, n-k];
Table[Print["T[", n, ",", k, "] = ", T[n, k]]; T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2020 *)

A046202 Distinct numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 5, 20, 30, 60, 7, 42, 105, 140, 8, 56, 168, 280, 9, 72, 252, 504, 630, 10, 90, 360, 840, 1260, 11, 110, 495, 1320, 2310, 2772, 132, 660, 1980, 3960, 5544, 13, 156, 858, 2860, 6435, 10296, 12012, 14, 182, 1092, 4004, 10010, 18018, 24024, 15

Offset: 1

Mohammad K. Azarian

Numbers in the order in which they appear in Leibniz's Harmonic Triangle (A003506). This sequence is a permutation of the natural numbers. - Robert G. Wilson v, Jun 12 2014

			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.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A003506, A046201, A014631.

t[n_, k_] := Denominator[n!*k!/(n + k + 1)!]; DeleteDuplicates@ Flatten@ Table[t[n - k, k], {n, 0, 14}, {k, 0, n/2}] (* Robert G. Wilson v, Jun 12 2014 *)

More terms from James Sellers, Dec 13 1999

A014764 Squares of distinct elements in Pascal triangle.

Original entry on oeis.org

1, 4, 9, 16, 36, 25, 100, 225, 400, 49, 441, 1225, 64, 784, 3136, 4900, 81, 1296, 7056, 15876, 2025, 14400, 44100, 63504, 121, 3025, 27225, 108900, 213444, 144, 4356, 48400, 245025, 627264, 853776, 169, 6084, 81796, 511225, 1656369, 2944656, 196

Offset: 1

Mohammad K. Azarian

Cf. A014631.

a(n) = A014631(n)^2. - Sean A. Irvine, Nov 19 2018

More terms from James Sellers

Offset changed by Sean A. Irvine, Nov 19 2018

cp's OEIS Frontend

A265912 Smallest m such that A014631(n) occurs in row m of Pascal's triangle.

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A119629 Inverse permutation to sequence A014631.

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A119741 A008279, with the first and last of each row removed.

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A132311 Triangle read by rows: T(n,k) is the number of partitions of binomial(n,k) into parts of the first n rows of Pascal's triangle, 0<=k<=n.

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A132312 Triangle read by rows: T(n,k) = number of partitions of binomial(n,k) into distinct parts of the first n rows of Pascal's triangle, 0<=k<=n.

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A046202 Distinct numbers in the triangle of denominators in Leibniz's Harmonic Triangle.

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Programs

Mathematica

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A014764 Squares of distinct elements in Pascal triangle.

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