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Irregular table read by rows; for n > 1, the n-th row lists the numbers of the form binomial(m+1, k) such that binomial(m, k) = n.
3

%I #6 Feb 16 2023 05:07:02

%S 3,4,6,5,10,6,15,7,10,21,8,28,9,36,10,45,11,15,20,55,12,66,13,78,14,

%T 91,15,105,16,21,35,120,17,136,18,153,19,171,20,190,21,35,210,22,28,

%U 56,231,23,253,24,276,25,300,26,325,27,351,28,378,29,36,84,406,30,435

%N Irregular table read by rows; for n > 1, the n-th row lists the numbers of the form binomial(m+1, k) such that binomial(m, k) = n.

%C In other words, the n-th rows lists the numbers that appear directly below n in Pascal's triangle (A007318).

%C The n-th row starts with n+1.

%H Rémy Sigrist, <a href="/A360655/a360655.gp.txt">PARI program</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%e Table begins:

%e n n-th row

%e -- --------------

%e 2 3

%e 3 4, 6

%e 4 5, 10

%e 5 6, 15

%e 6 7, 10, 21

%e 7 8, 28

%e 8 9, 36

%e 9 10, 45

%e 10 11, 15, 20, 55

%e 11 12, 66

%e 12 13, 78

%e .

%e For n = 6:

%e Pascal's triangle begins:

%e 1

%e 1 1

%e 1 2 1

%e 1 3 3 1

%e 1 4 6 4 1

%e 1 5 10 10 5 1

%e 1 6 15 20 15 6 1

%e 1 7 21 35 35 21 7 1

%e the value 6 appears in row 3 above 10, and in row 6 above 7 and 21,

%e so the 6th row contains 7, 10 and 21.

%o (PARI) See Links section.

%Y Cf. A007318, A360653, A360654.

%K nonn,tabf

%O 2,1

%A _Rémy Sigrist_, Feb 15 2023