A038338 → basis A013611

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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,              2304,               2304;
  1:            13824,             82944,             165888,              110592;
  2:           331776,           2654208,            7962624,            10616832,             5308416;
  3:          7962624,          79626240,          318504960,           637009920,           637009920,            254803968;
  4:        191102976,        2293235712,        11466178560,         30576476160,         45864714240,          36691771392,          12230590464;
  5:       4586471424,       64210599936,       385263599616,       1284211998720,       2568423997440,        3082108796928,        2054739197952,         587068342272;
  6:     110075314176,     1761205026816,     12328435187712,      49313740750848,     123284351877120,      197254963003392,      197254963003392,      112717121716224,       28179280429056;
  7:    2641807540224,    47552535724032,    380420285792256,    1775294667030528,    5325884001091584,    10651768002183168,    14202357336244224,    12173449145352192,     6086724572676096,     1352605460594688;
  8:   63403380965376,  1268067619307520,  11412608573767680,   60867245726760960,  213035360043663360,   511284864104792064,   852141440174653440,   973875931628175360,   730406948721131520,   324625310542725120,    64925062108545024;
  9: 1521681143169024, 33476985149718528, 334769851497185280, 2008619108983111680, 8034476435932446720, 22496534020610850816, 44993068041221701632, 64275811487459573760, 64275811487459573760, 42850540991639715840, 17140216396655886336, 3116402981210161152;

The row polynomials satisfy P(n,x) = (24+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 2304*x + 2304*x^2.

P[0, x_] := 576 + 2304*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 48*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 + 2304*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 48*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-c462762e7768ae01
• 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-c462762e7768ae01
• 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-c462762e7768ae01

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-c462762e7768ae01

  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 (67)

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

• basis-map coefficients of A026729 → basis A013609
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038325 → basis A007318
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A038325 → basis A013611
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A013609 → D
• A085881 → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A344565 → basis A007318
• basis-map coefficients of A021012 → basis A013609
• A038221 → basis A013611
• A344565 → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A028326 → basis A013611
• A038208 → x ↦ x/(x+1)
• A120910 → D
• A178821 → basis A007318
• A038243 → basis A013610
• A038338 → basis A007318
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A356546 → basis A013611
• A135838 → basis A013611
• A178820 → basis A013611
• A178821 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013609
• A135838 → x ↦ x/(x+1)
• A156992 → basis A007318
• A085880 → x ↦ x/(x+1)
• A156992 → basis A013611
• A344565 → x ↦ x/(x+1)
• A038207 → D
• A120909 → basis A013610
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• basis-map coefficients of A026729, shifted → basis A013609
• A013609 → basis A013610
• A007318 → D → x ↦ x/(x+1)
• A085880 → x ↦ x+1
• A130123 → basis A013609
• A178822 → x ↦ x/(x+1)
• A027466 → basis A013611
• A120909 → D
• A178821 → basis A013611
• A178820 → x ↦ x/(x+1)
• A178822 → x ↦ x+1
• A085880 → basis A013611
• basis-map coefficients of A113214, shifted → basis A013611
• A196347 → basis A007318
• A038208 → basis A013611
• A178820 → basis A007318
• A196347 → basis A013611
• A159854 → x ↦ x/(x+1)
• A038330 → basis A013609
• A135838 → x ↦ x+1
• basis-map coefficients of A030528, shifted → basis A013609
• A028326 → x ↦ x/(x+1)
• A356546 → basis A007318
• A178822 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A028326 → basis A007318
• A038208 → basis A007318
• A136572 → basis A013609
• A085881 → basis A007318

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