A038329 → basis A013610 → basis A013609

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:              576,               864,               324;
  1:            13824,             31104,             23328,               5832;
  2:           331776,            995328,           1119744,             559872,             104976;
  3:          7962624,          29859840,          44789760,           33592320,           12597120,            1889568;
  4:        191102976,         859963392,        1612431360,         1612431360,          906992640,          272097792,           34012224;
  5:       4586471424,       24078974976,       54177693696,        67722117120,        50791587840,        22856214528,         5714053632,         612220032;
  6:     110075314176,      660451885056,     1733686198272,      2600529297408,      2437996216320,      1462797729792,       548549148672,      117546246144,       11019960576;
  7:    2641807540224,    17832200896512,    53496602689536,     93619054706688,    105321436545024,     78991077408768,     39495538704384,    12694994583552,     2380311484416,     198359290368;
  8:   63403380965376,   475525357240320,  1604898080686080,   3209796161372160,   4212857461800960,   3791571715620864,   2369732322263040,  1015599566684160,   285637378129920,   47606229688320,   3570467226624;
  9: 1521681143169024, 12553869431144448, 47077010366791680, 105923273325281280, 158884909987921920, 166829155487318016, 125121866615488512, 67029571401154560, 25136089275432960, 6284022318858240, 942603347828736, 64268410079232;

The row polynomials satisfy P(n,x) = (24+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 864*x + 324*x^2.

P[0, x_] := 576 + 864*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 18*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 + 864*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 18*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_A013610 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

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → basis A013610 → basis A007318
• A120909 → basis A013610 → basis A013609
• A178820 → basis A007318 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• A038335 → D
• basis-map coefficients of A112227 → basis A013610 → basis A007318
• A038330 → x ↦ x/(x+1)
• A013609 → basis A013609 → D
• A120910 → basis A013611 → reverse
• A038330 → basis A013610 → reverse
• A038334 → basis A013609 → basis A013610
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• A156992 → basis A013611 → basis A013609
• A038221 → basis A013611 → basis A013609
• A196347 → x ↦ x+1 → basis A013611
• A136572 → basis A013610 → basis A007318
• A038335 → reverse
• A038208 → basis A013611 → basis A013609
• A356546 → basis A013611 → basis A013609
• A013610 → basis A013611 → x ↦ x+1
• A130123 → basis A013609 → basis A013609
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• A120910 → basis A013611 → D
• A013609 → D → basis A013609
• A038243 → basis A013609 → x ↦ x/(x+1)
• A196347 → basis A013611 → basis A013609
• A135065 → x ↦ x+1 → basis A013611
• A038330 → basis A013610 → D
• A038279 → basis A013610 → x ↦ x/(x+1)
• A027465 → D → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
• A130123 → basis A013610 → x ↦ x+1
• A028326 → basis A007318 → basis A013611
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• A028326 → basis A013611 → basis A013609
• A085881 → basis A013611 → basis A013609
• A038329 → basis A013611 → basis A013610
• A085880 → x ↦ x+1 → basis A013611
• A013610 → basis A007318 → D
• A003506 → x ↦ x+1 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
• A178822 → x ↦ x+1 → basis A013611
• A085880 → basis A013611 → basis A013609
• A178821 → basis A007318 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A013609 → basis A013609
• A027465 → x ↦ x/(x+1)
• A178821 → basis A013611 → basis A013609
• A013610 → basis A007318
• A178820 → basis A013611 → basis A013609
• A094305 → x ↦ x+1 → basis A013611
• A120910 → D → basis A013611
• A038284 → D
• A038330 → basis A013610
• A356546 → x ↦ x+1 → basis A013611
• A178822 → basis A013611 → basis A013609
• A038284 → reverse
• A120910 → basis A013611

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.