A038338 → basis A007318

OEIS matches ?

• A009980: 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:              576,              576,               144;
  1:            13824,            20736,             10368,              1728;
  2:           331776,           663552,            497664,            165888,             20736;
  3:          7962624,         19906560,          19906560,           9953280,           2488320,            248832;
  4:        191102976,        573308928,         716636160,         477757440,         179159040,          35831808,           2985984;
  5:       4586471424,      16052649984,       24078974976,       20065812480,       10032906240,        3009871872,         501645312,         35831808;
  6:     110075314176,     440301256704,      770527199232,      770527199232,      481579499520,      192631799808,       48157949952,       6879707136,       429981696;
  7:    2641807540224,   11888133931008,    23776267862016,    27738979172352,    20804234379264,    10402117189632,     3467372396544,     743008370688,     92876046336,      5159780352;
  8:   63403380965376,  317016904826880,   713288035860480,   951050714480640,   832169375170560,   499301625102336,   208042343792640,   59440669655040,  11145125560320,   1238347284480,    61917364224;
  9: 1521681143169024, 8369246287429632, 20923115718574080, 31384673577861120, 31384673577861120, 21969271504502784, 10984635752251392, 3923084197232640, 980771049308160, 163461841551360, 16346184155136, 743008370688;

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

P[0, x_] := 576 + 576*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 12*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 + 576*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 12*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-a2ff444bbbdc7b83
• 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-a2ff444bbbdc7b83
• 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-a2ff444bbbdc7b83

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 A007318 e-a2ff444bbbdc7b83

  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.

• A120909 → basis A013610
• A178821 → basis A007318
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• basis-map coefficients of A026729 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609
• A085880 → x ↦ x/(x+1)
• A178822 → x ↦ x+1
• A135838 → basis A013611
• A178820 → basis A007318
• A136572 → basis A013609
• A120909 → D
• A028326 → basis A013611
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A038208 → basis A007318
• A356546 → basis A007318
• A038330 → basis A013609
• A196347 → basis A013611
• A085881 → basis A007318
• A135838 → x ↦ x+1
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A007318 → D → x ↦ x/(x+1)
• A120910 → D
• A038208 → x ↦ x/(x+1)
• 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…
• A013609 → basis A013610
• A038208 → basis A013611
• A159854 → x ↦ x/(x+1)
• A196347 → basis A007318
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A038338 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609
• A344565 → basis A013611
• A028326 → basis A007318
• A344565 → basis A007318
• A156992 → basis A013611
• A085881 → basis A013611
• A178822 → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038207 → D
• basis-map coefficients of A113214, shifted → x ↦ x+1
• basis-map coefficients of A021012 → basis A013609
• A027466 → basis A013611
• A085880 → x ↦ x+1
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• basis-map coefficients of A113214, shifted → basis A013611
• A038325 → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A356546 → basis A013611
• A085880 → basis A013611
• A178822 → x ↦ x/(x+1)
• A038243 → basis A013610
• A130123 → basis A013609
• A178820 → basis A013611
• A178820 → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A344565 → x ↦ x/(x+1)
• A038221 → basis A013611
• A178821 → x ↦ x/(x+1)
• A178821 → basis A013611
• A038325 → basis A007318
• A013609 → D
• basis-map coefficients of A112227 → basis A013609
• A028326 → x ↦ 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-…
• A156992 → basis A007318
• A135838 → x ↦ x/(x+1)

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