A038338 → basis A008277

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OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:              576,                720,                 144;
  1:            13824,              32832,               15552,                1728;
  2:           331776,            1347840,             1140480,              290304,               20736;
  3:          7962624,           52503552,            70917120,            31104000,             4976640,              248832;
  4:        191102976,         1985679360,          4034064384,          2717245440,           731566080,            80621568,             2985984;
  5:       4586471424,        73777692672,        217463242752,        211443499008,         85279703040,         15551004672,          1254113280,           35831808;
  6:     110075314176,      2711034593280,      11323567964160,      15296168853504,       8677460606976,       2329640828928,        307006930944,        18919194624,         429981696;
  7:    2641807540224,     98918149128192,     576063677399040,    1053652946780160,     808331189944320,     299819356913664,      57428355317760,      5727356190720,      278628139008,       5159780352;
  8:   63403380965376,   3592755059097600,   28838074304692224,   70131940935598080,   70843681037352960,   34884800260079616,    9110954393468928,   1307694732410880,   102163650969600,    4024628674560,    61917364224;
  9: 1521681143169024, 130100022699098112, 1427340627334397952, 4553973347792191488, 5942328325916590080, 3780447394294923264, 1293269224893972480, 250562483822002176, 27951974905282560, 1757214796677120, 57211644542976, 743008370688;

The row polynomials satisfy P(n,x) = (24+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 576 + 720*x + 144*x^2.

P[0, x_] := 576 + 720*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 12*x)*P[n - 1, x] + 12*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 720*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 12*x)*v[i - 1] + 12*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-44f842c6e5783c9b
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-44f842c6e5783c9b
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-44f842c6e5783c9b

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A008277 e-44f842c6e5783c9b

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-e4b4d500ff341474 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-969ef99525bc6c6a root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (19)

Equal up to content, sign, reversal and a small index shift.

• A038325 → basis A008277
• A085881 → basis A008277
• A178822 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A135838 → basis A008277
• A344565 → basis A008277
• A038221 → basis A008277
• A126351 → reverse
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A178820 → basis A008277
• A085880 → basis A008277
• A178821 → basis A008277
• A028326 → basis A008277
• A038208 → basis A008277
• A143494 → reverse
• A156992 → basis A008277
• A196347 → basis A008277
• A356546 → basis A008277

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741