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 A136645 Triangle of coefficients of a Pascal sum of recursive orthogonal Hermite polynomials given in Hochstadt's book: P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}]. 0
 1, 1, 1, -1, 2, 1, -5, -2, 3, 1, -3, -16, -3, 4, 1, 21, -12, -35, -4, 5, 1, 43, 104, -33, -64, -5, 6, 1, -97, 246, 315, -74, -105, -6, 7, 1, -455, -656, 859, 752, -145, -160, -7, 8, 1, 361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1, 4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are: {1, 2, 2, -3, -17, -24, 52, 287, 197, -2202, -6674}; The idea here is that the Pascal triangle Binomial heights in the limit give a very normal/ Gaussian-like curve, so that these sums would, in the limit of large n as this linear sum, be more Hermite than other linear sums. The x^0 constants are, first column: {1, 1, -1, -5, -3, 21, 43, -97, -455, 361, 4951} REFERENCES Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, pp. 8, 42-43. LINKS Table of n, a(n) for n=1..66. FORMULA P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}]. EXAMPLE {1}, {1, 1}, {-1, 2, 1}, {-5, -2, 3, 1}, {-3, -16, -3, 4, 1}, {21, -12, -35, -4, 5, 1}, {43, 104, -33, -64, -5, 6, 1}, {-97, 246, 315, -74, -105, -6, 7,1}, {-455, -656, 859, 752, -145, -160, -7, 8, 1}, {361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1}, {4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1} MATHEMATICA P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; P2[x_, n_] := P2[x, n] = Sum[Binomial[n, m]*P[x, m], {m, 0, n}]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P2[x, n], x], {n, 0, 10}]; Flatten[a] CROSSREFS Cf. A137286. Sequence in context: A136262 A162180 A090003 * A366801 A247498 A091381 Adjacent sequences: A136642 A136643 A136644 * A136646 A136647 A136648 KEYWORD uned,tabl,sign AUTHOR Roger L. Bagula, Apr 01 2008 STATUS approved

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Last modified April 24 14:54 EDT 2024. Contains 371960 sequences. (Running on oeis4.)