A038328 → basis A013609 → basis A013610

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Submission text ?

Triangle begins:
  0:            324,             432,              144;
  1:           5832,           11664,             7776,             1728;
  2:         104976,          279936,           279936,           124416,            20736;
  3:        1889568,         6298560,          8398080,          5598720,          1866240,           248832;
  4:       34012224,       136048896,        226748160,        201553920,        100776960,         26873856,          2985984;
  5:      612220032,      2857026816,       5714053632,       6348948480,       4232632320,       1693052928,        376233984,         35831808;
  6:    11019960576,     58773123072,     137137287168,     182849716224,     152374763520,      81266540544,      27088846848,       5159780352,       429981696;
  7:   198359290368,   1190155742208,    3173748645888,    4936942338048,    4936942338048,    3291294892032,    1462797729792,     417942208512,     69657034752,     5159780352;
  8:  3570467226624,  23803114844160,   71409344532480,  126949945835520,  148108270141440,  118486616113152,   65825897840640,   25076532510720,   6269133127680,   928760463360,    61917364224;
  9: 64268410079232, 471301673914368, 1571005579714560, 3142011159429120, 4189348212572160, 3910058331734016, 2606705554489344, 1241288359280640, 413762786426880, 91947285872640, 12259638116352, 743008370688;

The row polynomials satisfy P(n,x) = (18+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 432*x + 144*x^2.

P[0, x_] := 324 + 432*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 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] = 324 + 432*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 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 (−∞, 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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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

Related ?

Same fingerprint ?

• A178822 → basis A013609 → basis A013610
• A085880 → basis A013611 → x ↦ x+1
• A013620 → D
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A132200 → basis A013610
• A028326 → basis A007318 → basis A013609
• A038334 → reverse
• A085881 → basis A013611 → basis A007318
• A013610 → D → basis A013609
• A120910 → basis A013609 → reverse
• A038220 → D
• A178821 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A038338 → basis A013610 → reverse
• A085880 → basis A013609 → basis A013610
• A135089 → basis A013610
• A027466 → basis A013611 → x ↦ x/(x+1)
• A178820 → basis A013611 → basis A007318
• A196347 → basis A013611 → x ↦ x+1
• A135838 → basis A013610
• A156992 → basis A013610
• A178821 → basis A013610 → D
• A135065 → basis A013610
• A038221 → basis A013610 → D
• A038243 → basis A013611 → D
• A356546 → x ↦ x+1 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A038221 → basis A013611 → x ↦ x+1
• A178820 → basis A013610 → D
• A003506 → x ↦ x+1 → basis A013609
• A038330 → basis A013609 → basis A013609
• A038243 → basis A013611
• A027465 → basis A013611 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A094305 → basis A013610
• A120910 → D → basis A013609
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038338 → basis A013610 → D
• A038208 → D → basis A013610
• A196347 → basis A013610
• A085880 → basis A013610 → reverse
• A085881 → basis A007318 → basis A013609
• A038221 → basis A013610
• A038325 → basis A013609 → basis A013610
• A356546 → basis A013610
• A178822 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A038221 → basis A013609 → basis A013610
• A038334 → D
• A178822 → basis A013611 → basis A007318
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A120910 → basis A013609
• A156992 → basis A013611 → x ↦ x+1
• A376827 → basis A013610
• A216973 → basis A013610
• A134059 → basis A013610
• A178822 → basis A013610
• A178821 → basis A013611 → basis A007318
• A085880 → basis A013610
• A085881 → basis A013609 → basis A013610
• A094305 → x ↦ x+1 → basis A013609
• A038330 → basis A013611 → basis A013610
• A028326 → basis A013611 → x ↦ x+1
• basis-map coefficients of A113214, shifted → D → basis A013610
• A196347 → basis A013610 → D
• A085881 → basis A013610
• A136572 → basis A013609 → x ↦ x+1
• A344565 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A178820 → basis A013609 → basis A013610
• A178820 → basis A007318 → basis A013609
• A013610 → basis A013611 → basis A013609
• A134058 → basis A013610
• A156992 → basis A013609 → basis A013610
• A038325 → basis A007318 → basis A013609
• A178821 → basis A013610 → reverse
• A356546 → basis A013609 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A136572 → basis A013610 → basis A013609
• A038325 → basis A013610
• A038208 → basis A013610 → D
• A134346 → basis A013610
• A028326 → basis A013610 → D
• basis-map coefficients of A113214, shifted → basis A013610
• A038208 → basis A013610
• A120910 → basis A013609 → D
• A356546 → basis A013611 → basis A007318
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A003506 → D → basis A013610
• A156992 → basis A007318 → basis A013609
• A258758 → basis A013610
• A178820 → basis A013610
• A085880 → basis A013610 → D
• A178822 → x ↦ x+1 → basis A013609
• A028326 → D → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A038330 → basis A013609 → x ↦ x/(x+1)
• A131431 → x ↦ x+1 → basis A013610
• A028326 → basis A013609 → basis A013610
• A013609 → basis A013610 → x ↦ x+1
• A124860 → basis A013610
• A038291 → basis A013609 → basis A013611
• A178821 → basis A013609 → basis A013610
• A120910 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• A038338 → basis A013610
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• A028326 → basis A013610
• A085880 → D → basis A013610
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A156992 → basis A013610 → D
• A135065 → x ↦ x+1 → basis A013609
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A038338 → basis A013609 → basis A013610
• basis-map coefficients of A127647 → basis A013610
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A120910 → basis A013610 → basis A013610

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.