A038337 → basis A013609

OEIS matches ?

• A009989: row sums, 15 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:             529,              1012,               484;
  1:           12167,             34914,             33396,              10648;
  2:          279841,           1070696,           1536216,             979616,             234256;
  3:         6436343,          30782510,          58888280,           56327920,           26939440,            5153632;
  4:       148035889,         849597276,        2031645660,         2591084320,         1858821360,          711201216,          113379904;
  5:      3404825447,       22797526906,       65418990252,       104291143880,        99756746320,        57251697888,        18254164544,         2494357888;
  6:     78310985281,      599249278672,     2006182367728,      3837914094784,      4588810330720,      3511437470464,      1679383138048,       458961851392,        54875873536;
  7:   1801152661463,    15505575085638,    59325678588528,    132408036270048,    189976747691808,    181716889096512,    115877436525312,     47502551619072,     11359305821952,     1207269217792;
  8:  41426511213649,   396253585521860,  1705613259420180,   4350549763158720,   7282441994852640,   8358976898439552,   6662952600205440,   3641862290795520,   1306320169524480,   277671920092160,   26559922791424;
  9: 952809757913927, 10025215713703058, 47946683848145060, 137586136259894520, 263208260671102560, 352470192550867776, 337145401570395264, 230347789892816640, 110166334296564480, 35125497891658240, 6719660466230272, 584318301411328;

The row polynomials satisfy P(n,x) = (23+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 1012*x + 484*x^2.

P[0, x_] := 529 + 1012*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 22*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 529 + 1012*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 22*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-9947ebaa7f991114
• 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-9947ebaa7f991114
• 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-9947ebaa7f991114

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 A013609 e-9947ebaa7f991114

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-37ab130d55b994ea root: theorem check

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

    • basis_map A007318 e-107a218257e6f127

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (1+11x)^n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-5d158048399db1f0 root: theorem check

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

        • parent_provers periodic_product e-70681fa17f5604ff root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-9aedabfb67a4638b root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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