A013618 → basis A013611

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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:          144,           1056,            1936;
  1:         1728,          19008,           69696,            85184;
  2:        20736,         304128,         1672704,          4088832,           3748096;
  3:       248832,        4561920,        33454080,        122664960,         224885760,          164916224;
  4:      2985984,       65691648,       602173440,       2943959040,        8095887360,        11873968128,         7256313856;
  5:     35831808,      919683072,     10116513792,      61823139840,      226684846080,       498706661376,       609530363904,        319277809664;
  6:    429981696,    12612796416,    161864220672,    1187004284928,     5440436305920,     15958613164032,     29257457467392,      30650669727744,      14048223625216;
  7:   5159780352,   170272751616,   2497333690368,   21366077128704,   117513424207872,    430882555428864,   1053268468826112,    1655136165298176,    1517208151523328,     618121839509504;
  8:  61917364224,  2270303354880,  37460005355520,  366275607920640,  2350268484157440,  10341181330292736,  31598054064783360,   66205446611927040,   91032489091399680,   74174620741140480,   27197360938418176;
  9: 743008370688, 29968004284416, 549413411880960, 6043547530690560, 44319348558397440, 227505989266440192, 834188627310280704, 2184779738193592320, 4005429520021585920, 4895524968915271680, 3590051643871199232, 1196683881290399744;

The row polynomials satisfy P(n,x) = (12+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1056*x + 1936*x^2.

P[0, x_] := 144 + 1056*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 44*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 1056*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 44*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-f283d143fa0950f0
• 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-f283d143fa0950f0
• 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-f283d143fa0950f0

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 A013611 e-f283d143fa0950f0

  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 ?

Related sequences ?

Same fingerprint (2)

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

• A038281 → basis A007318
• A038303 → 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