User talk:Adi Dani

In my contributions here Set partitions and Set compositions I establish following tables

1

m\k 0 1 2 3 4 5 6 7 8 9 10 .. ${\overline {p}}_{m}(N_{2})$ 0 1 0 0 0 0 0 0 0 0 0 0 . 1 1 0 1 1 0 0 0 0 0 0 0 0 . 2 2 0 0 1 3 3 0 0 0 0 0 0 . 7 3 0 0 0 1 6 15 15 0 0 0 0 . 37 4 0 0 0 0 1 10 45 105 105 0 0 . 266 5 0 0 0 0 0 1 15 105 420 945 945 . 2431 .. . . . . . . . . . . . . . ${\overline {p}}(k,N_{2})$ 1 1 2 4 10 26 76 . . 2 m\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .. ${\overline {p}}_{m}(N_{3})$ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 . 3 2 0 0 1 3 7 10 10 0 0 0 0 0 0 0 0 0 . 31 3 0 0 0 1 6 25 75 175 280 280 0 0 0 0 0 0 . 842 4 0 0 0 0 1 10 65 315 1225 3780 9100 15400 15400 0 0 0 . 45296 5 0 0 0 0 0 1 15 140 980 5565 26145 102025 323400 800800 1401400 1401400 . 4061871 .. . . . . . . . . . . . . . ${\overline {p}}(k,N_{3})$ 1 1 2 5 14 46 . . . 3 m\k 0 1 2 3 4 5 6 7 8 9 10 .. ${\overline {c}}_{m}(N_{2})$ 0 1 0 0 0 0 0 0 0 0 0 0 . 1 1 0 1 1 0 0 0 0 0 0 0 0 . 2 2 0 0 2 6 6 0 0 0 0 0 0 . 14 3 0 0 0 6 36 90 90 0 0 0 0 . 222 4 0 0 0 0 24 240 1080 2520 2520 0 0 . 6384 5 0 0 0 0 0 120 1800 12600 50400 113400 113400 . 291720 .. . . . . . . . . . . . . . ${\overline {c}}(k,N_{2})$ 1 1 3 12 66 450 3690 . . 4 This table is proposed to bi part of OEIS A189804--Adi Dani 21:29, 27 April 2011 (UTC) m\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .. ${\overline {c}}_{m}(N_{3})$ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 . 3 2 0 0 2 6 14 20 20 0 0 0 0 0 0 0 0 0 . 62 3 0 0 0 6 36 150 450 1050 1680 1680 0 0 0 0 0 0 . 5052 4 0 0 0 0 24 240 1560 7560 29400 90720 218400 369600 369600 0 0 0 . 10871804 5 0 0 0 0 0 120 1800 16800 117600 667880 3137400 12243000 38880800 96096000 168168000 168168000 . 487424520 .. . . . . . . . . . . . . . ${\overline {c}}(k,N_{3})$ 1 1 3 13 74 530 4550 . . My sequence Take a look at the last column of the tables above we can see that: ${\overline {p}}_{m}(N_{2}):1,2,7,37,266,2431,...$-A001515 ${\overline {p}}_{m}(N_{3}):1,3,31,842,45296,4061871,...$-A144416 ${\overline {c}}_{m}(N_{2}):1,2,14,222,6384,291720,...$that is A105749 ${\overline {c}}_{m}(N_{3}):1,3,62,5052,1087104,...$ that is A144422 and if we show the last row of the tables we can see that: ${\overline {p}}(k,N_{2}):1,1,2,4,10,26,76,...$-A000085 ${\overline {p}}(k,N_{3}):1,1,2,5,14,46,166,...$-A001680 ${\overline {c}}(k,N_{2}):1,1,3,12,66,450,3690,...$-A080599 ${\overline {c}}(k,N_{3}):1,1,3,13,74,530,4550,...$ From [1] we can see that last sequence is not in database of OEIS I think that before proclaiming the authorship for this sequence I needs help and instruction step by step from administrators. Hello Adi, nice work! Since you draw my attention to this page on my user talk page I will try to give you some hints how you can proceed next (though notice that I am not an administrator or an editor). Since you have already all the important pieces gathered (recurrence, generating function, etc.) you just have to bring them into the format required for a database entry. The submission form looks similar to this form. This page is a dummy only but it gives you the opportunity to consider the submission items as they are required. So you might write up the entries in advance (they can later be filled into the real submission form by copy and paste). After you have written such a draft you can show it here if you want further comments before you submit. Peter Luschny 11:52, 25 April 2011 (UTC) Mathematica formula Table[Sum[k!m!/(2^(k+j-2m)3^(m-j)(m-j)!(k+2j-3m)!(3m-j-k)!),{m,0,k},{j,0,3m-k}],{k,0,20}] A078012 A078012 Sequence of the Day for a day in September? Would you like to choose the Template:Sequence of the Day for September 6 and write a couple of words about it? Alonso del Arte 00:33, 31 May 2011 (UTC) Notation for sets In User:Adi Dani#Note on notations, you have the set identified with italic capital letters (1)......$N=\{0,1,2,...\}\,$ the set of natural numbers (2)......$I_{a}^{b}=\{x:b\leq x<a,x\in N\}\,$ (3)......$I_{a}^{0}=I_{a}\,$ (4)......$I_{a+1}^{1}=N_{a}\,$ (5)......$I_{\infty }^{b}=I^{b}\,$ (6)......$O=\{x:x=2n+1,n\in N\}\,$ (7)......$E=\{x:x=2n,n\in N\}\,$ :(1)......<math>N=\{0,1,2,...\}\,</math> the set of natural numbers :(2)......<math>I_{a}^{b}=\{x:b\le x<a, x\in N\}\,</math> :(3)......<math>I_{a}^{0}=I_{a}\,</math> :(4)......<math>I_{a+1}^{1}=N_{a} \,</math> :(5)......<math>I_{\infty}^{b}=I^{b} \,</math> :(6)......<math>O=\{x: x=2n+1,n\in N\}\,</math> :(7)......<math>E=\{x: x=2n, n\in N\}\,</math> although the usual convention is not to use italic letters (and to use blackboard, i.e. double stroke, letters for many standard sets) $\mathbb {N} =\{0,1,2,\ldots \}\,$ the set of natural numbers ${\rm {I}}_{a}^{b}=\{x:b\leq x<a,x\in \mathbb {N} \}\,$ ${\rm {I}}_{a}^{0}={\rm {I}}_{a}\,$ ${\rm {I}}_{a+1}^{1}=\mathbb {N} _{a}\,$ ${\rm {I}}_{\infty }^{b}={\rm {I}}^{b}\,$ $\mathbb {O} =\{x:x=2n+1,n\in \mathbb {N} \}\,$ $\mathbb {E} =\{x:x=2n,n\in \mathbb {N} \}\,$ # <math>\mathbb{N} = \{ 0, 1, 2, \ldots \} \,</math> the set of [[natural numbers]] # <math>{\rm I}_{a}^{b} = \{ x: b \le x < a, x \in \mathbb{N} \} \,</math> # <math>{\rm I}_{a}^{0} = {\rm I}_{a} \,</math> # <math>{\rm I}_{a+1}^{1} = \mathbb{N}_{a} \,</math> # <math>{\rm I}_{\infty}^{b} = {\rm I}^{b} \,</math> # <math>\mathbb{O} = \{ x: x = 2n+1, n \in \mathbb{N} \} \,</math> # <math>\mathbb{E} = \{ x: x = 2n, n \in \mathbb{N} \} \,</math> — Daniel Forgues 01:54, 15 June 2011 (UTC)