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A140880
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Coefficients of the second derivative of the binomial / Pascal's triangle polynomials plus second integer differential on the n column direction: D02(n,m)=Coefficients(d^2/dx^2((1+x)^n); D20=(n, m)= Binomial(n + 2, m) - 2*Binomial(n + 1, m) - Binomial(n, m); t(n,m)=D02(n,m)+D20(n,m).
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0
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2, 6, 6, 12, 24, 12, 20, 60, 60, 20, 30, 120, 180, 120, 30, 42, 210, 420, 420, 210, 42, 56, 336, 840, 1120, 840, 336, 56, 72, 504, 1512, 2520, 2520, 1512, 504, 72, 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Row sums are: {0, 0, 0, 4, 27, 114, 388, 1168, 3253, 8594, 21862, ...}.
This sequence is an attempt to get a two-variable-like differentiation of Pascal's triangle.
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FORMULA
| D02(n,m)=Coefficients(d^2/dx^2((1+x)^n); D20=(n, m)= Binomial(n + 2, m) - 2*Binomial(n + 1, m) - Binomial(n, m); t(n,m)=D02(n,m)+D20(n,m).
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EXAMPLE
| {0},
{0},
{0},
{4, 0},
{10, 16, 1},
{18, 50, 41, 5},
{28, 108, 151, 86, 15},
{40, 196, 379, 357, 161, 35},
{54, 320, 785, 1016, 728, 280, 70},
{70, 486, 1441, 2361, 2304, 1344, 462, 126},
{88, 700, 2431, 4810, 5925, 4656, 2310, 732, 210}
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MATHEMATICA
| Clear[D2, D02, D20, a, n, m, x] T[n_, m_] := Binomial[n, m]; D02[n_, m_] := CoefficientList[D[(1 + x)^n, {x, 2}], x][[m + 1]]; D20[n_, m_] := T[n + 2, m] - 2*T[n + 1, m] - T[n, m]; D2[n_, m_] := D02[n, m] + D20[n, m]; a = Table[Table[D2[n, m], {m, 0, n - 2}], {n, 0, 10}]; Flatten[a]
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CROSSREFS
| Cf. A007318.
Sequence in context: A064797 A053319 A075779 * A065420 A119312 A051398
Adjacent sequences: A140877 A140878 A140879 * A140881 A140882 A140883
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KEYWORD
| nonn,uned,tabl
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jul 22 2008
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