A038329 → x ↦ x+1 → basis A013609

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Submission text ?

Triangle begins:
  0:            324,             216,              36;
  1:           5832,            5832,            1944,             216;
  2:         104976,          139968,           69984,           15552,            1296;
  3:        1889568,         3149280,         2099520,          699840,          116640,            7776;
  4:       34012224,        68024448,        56687040,        25194240,         6298560,          839808,          46656;
  5:      612220032,      1428513408,      1428513408,       793618560,       264539520,        52907904,        5878656,        279936;
  6:    11019960576,     29386561536,     34284321792,     22856214528,      9523422720,      2539579392,      423263232,      40310784,       1679616;
  7:   198359290368,    595077871104,    793437161472,    617117792256,    308558896128,    102852965376,    22856214528,    3265173504,     272097792,     10077696;
  8:  3570467226624,  11901557422080,  17852336133120,  15868743229440,   9256766883840,   3702706753536,  1028529653760,  195910410240,   24488801280,   1813985280,    60466176;
  9: 64268410079232, 235650836957184, 392751394928640, 392751394928640, 261834263285760, 122189322866688, 40729774288896, 9697565306880, 1616260884480, 179584542720, 11972302848, 362797056;

The row polynomials satisfy P(n,x) = (18+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 216*x + 36*x^2.

P[0, x_] := 324 + 216*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 216*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 6*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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A130123 → D → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
• A131431 → D → basis A013610
• basis-map coefficients of A112227 → basis A013610
• A131431 → θ+1 → basis A013610
• A127648 → D → basis A013610
• A038338 → x ↦ x+1 → basis A007318
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A038221 → basis A013611 → x ↦ x/(x+1)
• A136572 → (θ+1)² → basis A013610
• basis-map coefficients of A112227 → basis A013610 → reverse
• A129186 → basis A013610
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
• A136572 → basis A013610
• A356546 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A112227 → (θ+1)² → basis A013610
• A013610 → D → basis A013611
• A013609 → basis A013610 → x ↦ x/(x+1)
• A127648 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
• A156992 → basis A013611 → x ↦ x/(x+1)
• A038243 → basis A013609
• A136572 → D → basis A013610
• A013610 → basis A013611
• A085880 → basis A013611 → x ↦ x/(x+1)
• A130123 → basis A013610 → basis A013611
• basis-map coefficients of A026729 → basis A013610 → D
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• basis-map coefficients of A026729, shifted → θ+1 → basis A013610
• A038330 → basis A013609 → x ↦ x+1
• basis-map coefficients of A026729 → (θ+1)² → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013611
• A132440 → θ+1 → basis A013610
• A013610 → D
• basis-map coefficients of A112227 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → D
• A027465 → D
• A027465 → D → D
• A135065 → x ↦ x+1 → basis A007318
• A131436 → basis A013610
• A356546 → basis A013611 → x ↦ x/(x+1)
• A130123 → basis A013610 → D
• A085881 → basis A007318 → basis A007318
• A136572 → θ+1 → basis A013610
• A196347 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013610 → D
• A038279 → basis A013610 → D
• A130123 → (θ+1)² → basis A013610
• basis-map coefficients of A026729 → basis A013610 → reverse
• A013610 → basis A013611 → reverse
• A178821 → basis A007318 → x ↦ x+1
• A178821 → basis A013611 → x ↦ x/(x+1)
• A038243 → basis A013609 → D
• basis-map coefficients of A021012 → θ+1 → basis A013610
• basis-map coefficients of A021012 → basis A013610
• A196347 → x ↦ x+1 → basis A007318
• A178820 → basis A013611 → x ↦ x/(x+1)
• A038330 → D
• image under θ+1 → basis A013610
• A136521 → basis A013610
• basis-map coefficients of A112227 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → θ+1 → basis A013610
• A085881 → basis A013611 → x ↦ x/(x+1)
• A147716 → basis A013610 → basis A013609
• A131431 → basis A013610
• A130123 → basis A013610
• A120910 → basis A007318 → reverse
• image under θ+1 → (θ+1)² → basis A013610
• A094305 → x ↦ x+1 → x ↦ x+1
• A178822 → x ↦ x+1 → x ↦ x+1
• image under θ+1 → θ+1 → D → basis A013610
• basis-map coefficients of A026729, shifted → D → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013611
• A085880 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A030528, shifted → basis A013610
• A028326 → basis A007318 → basis A007318
• A120909 → x ↦ x/(x+1)
• A013610 → D → D
• A134309 → basis A013610
• A028326 → basis A013611 → x ↦ x/(x+1)
• A038279 → basis A013610
• A156992 → basis A007318 → x ↦ x+1
• A130123 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → D → basis A013610
• A178822 → basis A013611 → x ↦ x/(x+1)
• A038330 → reverse
• A027465: Cube of lower triangular normalized binomial matrix.
• basis-map coefficients of A026729 → D → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610
• basis-map coefficients of A026729 → basis A013610
• A120910 → basis A007318
• A178820 → basis A007318 → basis A007318
• A038279 → basis A013610 → reverse
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
• A120909 → basis A013610 → x ↦ x/(x+1)
• A013610 → basis A013611 → D
• A136572 → basis A013610 → D
• A136572 → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
• A120910 → D → x ↦ x+1
• A038325 → basis A007318 → basis A007318

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.