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 A142240 A triangular sequence from the pattern in row sums of Pascal's triangle A007318, Eulerian numbers A008292 and A060187: Delta_diagonal=m; m={0,1,2,3,...k}. 0
 1, 2, 2, 2, 3, 2, 2, 4, 4, 2, 2, 5, 6, 5, 2, 2, 6, 8, 8, 6, 2, 2, 7, 10, 11, 10, 7, 2, 2, 8, 12, 14, 14, 12, 8, 2, 2, 9, 14, 17, 18, 17, 14, 9, 2, 2, 10, 16, 20, 22, 22, 20, 16, 10, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums are: {1, 4, 7, 12, 20, 32, 49, 72, 102, 140}. The triangle is calculated by hand. Row sums are: 1) Pascal A007318:ratio =2: delta row zero:a(n)=2*a(n-1);a(1)=1; b(n)->0 1,2,4,8,16,32,64,128,256,512 2) Eulerian numbers A008292: ratio =n: delta=1:a(n)=n*a(n-1) 1,2,6,24,120,720,...n! b(n)->1,2,3,4,... 3) A060187:ratio=2*n: delta=2:a(n)=2*n*a(n-1) b(n)->2,4,6,8,... {1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200}. 4) hypothetical next level sums:delta=3:b(n)=b(n-1)+3;a(n)=a(n-1)*b(n); b(n)->{2, 5, 8, 11, 14, 17, 20, 23, 26, 29} diagonal {1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400, 2504902400, 72642169600} 5)hypothetical next level sums:delta=3:b(n)=b(n-1)+4;a(n)=a(n-1)*b(n); b(n)->{2, 6, 10, 14, 18, 22, 26, 30, 34, 38} diagonal {1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800} The conjecture that goes with this triangular sequence is that there on n levels like these for Pascal combinatorial quantum levels. LINKS FORMULA b(n,m)=b(n-1,m]+m; Delta_diagonal=m; m={0,1,2,3,...k}. EXAMPLE {1}, {2, 2}, {2, 3, 2}, {2, 4, 4, 2}, {2, 5, 6, 5, 2}, {2, 6, 8, 8, 6, 2}, {2, 7, 10, 11, 10, 7, 2}, {2, 8, 12, 14, 14, 12, 8, 2}, {2, 9, 14, 17, 18, 17, 14, 9, 2}, {2, 10, 16, 20, 22, 22, 20, 16, 10, 2} MATHEMATICA a={{1}, {2, 2}, {2, 3, 2}, {2, 4, 4, 2}, {2, 5, 6, 5, 2}, {2, 6, 8, 8, 6, 2}, {2, 7, 10, 11, 10, 7, 2}, {2, 8, 12, 14, 14, 12, 8, 2}, {2, 9, 14, 17, 18, 17, 14, 9, 2}, {2, 10, 16, 20, 22, 22, 20, 16, 10, 2}} Flatten[a] Table[Apply[Plus, a[[n]]], {n, 1, 10}] CROSSREFS Cf. A060187, A007318, A008292. Sequence in context: A099774 A305973 A290978 * A284565 A227911 A325783 Adjacent sequences:  A142237 A142238 A142239 * A142241 A142242 A142243 KEYWORD nonn,uned AUTHOR Roger L. Bagula and Gary W. Adamson, Sep 17 2008 STATUS approved

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Last modified October 13 19:48 EDT 2019. Contains 327981 sequences. (Running on oeis4.)