A038338 → x ↦ x+1 → basis A007318

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Submission text ?

Triangle begins:
  0:               1296,                864,                144;
  1:              46656,              46656,              15552,               1728;
  2:            1679616,            2239488,            1119744,             248832,              20736;
  3:           60466176,          100776960,           67184640,           22394880,            3732480,             248832;
  4:         2176782336,         4353564672,         3627970560,         1612431360,          403107840,           53747712,           2985984;
  5:        78364164096,       182849716224,       182849716224,       101583175680,        33861058560,         6772211712,         752467968,          35831808;
  6:      2821109907456,      7522959753216,      8776786378752,      5851190919168,      2437996216320,       650132324352,      108355387392,       10319560704,        429981696;
  7:    101559956668416,    304679870005248,    406239826673664,    315964309635072,    157982154817536,     52660718272512,    11702381838336,     1671768834048,     139314069504,      5159780352;
  8:   3656158440062976,  12187194800209920,  18280792200314880,  16249593066946560,   9478929289052160,   3791571715620864,  1053214365450240,   200612260085760,   25076532510720,   1857520926720,    61917364224;
  9: 131621703842267136, 482612914088312832, 804354856813854720, 804354856813854720, 536236571209236480, 250243733230977024, 83414577743659008, 19860613748490240, 3310102291415040, 367789143490560, 24519276232704, 743008370688;

The row polynomials satisfy P(n,x) = (36+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 864*x + 144*x^2.

P[0, x_] := 1296 + 864*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 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] = 1296 + 864*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by 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)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

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

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

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A120910 → basis A007318 → reverse
• A131431 → D → basis A013610
• A131431 → θ+1 → basis A013610
• A127648 → D → basis A013610
• A038329 → x ↦ x+1 → basis A013609
• A136572 → (θ+1)² → basis A013610
• A178821 → basis A013611 → x ↦ x/(x+1)
• A013610 → D → D
• A038221 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013610 → D
• A129186 → basis A013610
• A038330 → basis A013609 → x ↦ x+1
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
• basis-map coefficients of A112227 → (θ+1)² → basis A013610
• A013610 → D → basis A013611
• A127648 → basis A013610
• A038330 → reverse
• A136572 → D → basis A013610
• A027465: Cube of lower triangular normalized binomial matrix.
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
• A130123 → basis A013610 → basis A013611
• basis-map coefficients of A021012 → basis A013610
• basis-map coefficients of A026729, shifted → θ+1 → basis A013610
• A094305 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A026729 → (θ+1)² → basis A013610
• A013610 → basis A013611
• basis-map coefficients of A112227 → basis A013610 → basis A013611
• A196347 → x ↦ x+1 → basis A007318
• A132440 → θ+1 → basis A013610
• A178820 → basis A007318 → basis A007318
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610
• A130123 → basis A013610
• A038330 → D
• A085880 → basis A013611 → x ↦ x/(x+1)
• A136572 → basis A013610
• A131436 → basis A013610
• A013610 → D
• A136572 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
• A178820 → basis A013611 → x ↦ x/(x+1)
• A013610 → basis A013611 → reverse
• A013609 → basis A013610 → x ↦ x/(x+1)
• A130123 → (θ+1)² → basis A013610
• A178821 → basis A007318 → x ↦ x+1
• A136572 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A013610 → D
• A085880 → x ↦ x+1 → x ↦ x+1
• A038325 → basis A007318 → basis A007318
• A120910 → D → x ↦ x+1
• A356546 → basis A013611 → x ↦ x/(x+1)
• A028326 → basis A013611 → x ↦ x/(x+1)
• A178822 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A021012 → θ+1 → basis A013610
• A135065 → x ↦ x+1 → basis A007318
• basis-map coefficients of A112227 → basis A013610 → reverse
• A147716 → basis A013610 → basis A013609
• basis-map coefficients of A026729 → basis A013610
• image under θ+1 → basis A013610
• A130123 → basis A013610 → D
• A027465 → D
• A136521 → basis A013610
• basis-map coefficients of A112227 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → θ+1 → basis A013610
• A038279 → basis A013610 → reverse
• A131431 → basis A013610
• A120910 → basis A007318
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A038279 → basis A013610
• image under θ+1 → (θ+1)² → basis A013610
• A356546 → 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
• A156992 → basis A007318 → x ↦ x+1
• A028326 → basis A007318 → basis A007318
• basis-map coefficients of A112227 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → reverse
• A134309 → basis A013610
• A130123 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → D → basis A013610
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A038243 → basis A013609
• basis-map coefficients of A026729 → D → basis A013610
• basis-map coefficients of A112227 → basis A013610 → D
• A196347 → basis A013611 → x ↦ x/(x+1)
• A013610 → basis A013611 → D
• A156992 → basis A013611 → x ↦ x/(x+1)
• A038243 → basis A013609 → D
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
• A178822 → x ↦ x+1 → x ↦ x+1
• A136572 → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → D
• A085881 → basis A013611 → x ↦ x/(x+1)
• A120909 → basis A013610 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013610
• A120909 → x ↦ x/(x+1)
• A027465 → D → D
• A038279 → basis A013610 → D
• A130123 → D → basis A013610
• A085881 → 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.