A038338 → basis A013610

OEIS matches ?

• A159991: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,              1728,               1296;
  1:            13824,             62208,              93312,              46656;
  2:           331776,           1990656,            4478976,            4478976,             1679616;
  3:          7962624,          59719680,          179159040,          268738560,           201553920,            60466176;
  4:        191102976,        1719926784,         6449725440,        12899450880,         14511882240,          8707129344,          2176782336;
  5:       4586471424,       48157949952,       216710774784,       541776936960,        812665405440,        731398864896,        365699432448,         78364164096;
  6:     110075314176,     1320903770112,      6934744793088,     20804234379264,      39007939461120,      46809527353344,      35107145515008,      15045919506432,       2821109907456;
  7:    2641807540224,    35664401793024,    213986410758144,    748952437653504,    1685142984720384,    2527714477080576,    2527714477080576,    1624959306694656,     609359740010496,     101559956668416;
  8:   63403380965376,   951050714480640,   6419592322744320,  25678369290977280,   67405719388815360,  121330294899867648,  151662868624834560,  129996744535572480,   73123168801259520,   24374389600419840,   3656158440062976;
  9: 1521681143169024, 25107738862288896, 188308041467166720, 847386186602250240, 2542158559806750720, 5338532975594176512, 8007799463391264768, 8579785139347783680, 6434838854510837760, 3217419427255418880, 965225828176625664, 131621703842267136;

The row polynomials satisfy P(n,x) = (24+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1728*x + 1296*x^2.

P[0, x_] := 576 + 1728*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 36*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 1728*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 36*x)*v[i - 1]); 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-85e8843cd1dca083
• 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-85e8843cd1dca083
• 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-85e8843cd1dca083

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 A013610 e-85e8843cd1dca083

  Geometric basis map (elementary) polytool ?

  • 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 (23)

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

• A038243 → basis A013611
• A178821 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A120910 → basis A013609
• A344565 → basis A013610
• A178822 → basis A013610
• A135838 → basis A013610
• A028326 → basis A013610
• A013620 → D
• A120910 → x ↦ x/(x+1)
• A038325 → basis A013610
• A196347 → basis A013610
• A038221 → basis A013610
• A085881 → basis A013610
• A038208 → basis A013610
• A038220 → D
• A085880 → basis A013610
• A156992 → basis A013610
• A356546 → basis A013610
• A178820 → basis A013610

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