A038330 → basis A013611 → basis A013610

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Submission text ?

Triangle begins:
  0:              1024,               3072,                2304;
  1:             32768,             147456,              221184,               110592;
  2:           1048576,            6291456,            14155776,             14155776,              5308416;
  3:          33554432,          251658240,           754974720,           1132462080,            849346560,             254803968;
  4:        1073741824,         9663676416,         36238786560,          72477573120,          81537269760,           48922361856,           12230590464;
  5:       34359738368,       360777252864,       1623497637888,        4058744094720,        6088116142080,         5479304527872,         2739652263936,          587068342272;
  6:     1099511627776,     13194139533312,      69269232549888,      207807697649664,      389639433093120,       467567319711744,       350675489783808,       150289495621632,        28179280429056;
  7:    35184372088832,    474989023199232,    2849934139195392,     9974769487183872,    22443231346163712,     33664847019245568,     33664847019245568,     21641687369515008,      8115632763568128,     1352605460594688;
  8:  1125899906842624,  16888498602639360,  113997365567815680,   455989462271262720,  1196972338462064640,   2154550209231716352,   2693187761539645440,   2308446652748267520,   1298501242170900480,   432833747390300160,    64925062108545024;
  9: 36028797018963968, 594475150812905472, 4458563631096791040, 20063536339935559680, 60190609019806679040, 126400278941594025984, 189600418412391038976, 203143305441847541760, 152357479081385656320, 76178739540692828160, 22853621862207848448, 3116402981210161152;

The row polynomials satisfy P(n,x) = (32+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 3072*x + 2304*x^2.

P[0, x_] := 1024 + 3072*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 48*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1024 + 3072*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 48*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • 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
            • 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
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A134059 → basis A013610
• A178822 → basis A013610
• A038338 → basis A013610 → reverse
• A344565 → basis A013610
• A178821 → basis A013610
• A178821 → basis A013609 → basis A013610
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• A028326 → basis A007318 → basis A013609
• A178822 → basis A013610 → D
• A178820 → basis A013610
• A120910 → x ↦ x/(x+1)
• A120910 → D → basis A013609
• A038334 → reverse
• A028326 → basis A013611 → x ↦ x+1
• A196347 → basis A013611 → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A156992 → basis A013610
• A258758 → basis A013610
• A028326 → basis A013610
• A038330 → basis A013609 → basis A013609
• A094305 → basis A013610
• A028326 → D → basis A013610
• A013610 → D → basis A013609
• A013609 → basis A013610 → x ↦ x+1
• A124860 → basis A013610
• A038330 → basis A013609 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A132200 → basis A013610
• A135089 → basis A013610
• A178822 → basis A013611 → basis A007318
• A038221 → basis A013609 → basis A013610
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• basis-map coefficients of A127647 → basis A013610
• A085881 → basis A013611 → basis A007318
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A085881 → basis A007318 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A356546 → basis A013611 → basis A007318
• basis-map coefficients of A113214, shifted → basis A013610
• A038243 → basis A013611 → D
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A120910 → basis A013609 → D
• A178821 → basis A013611 → basis A007318
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A038325 → basis A013610
• A376827 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A038338 → basis A013609 → basis A013610
• A120910 → basis A013609
• A135065 → x ↦ x+1 → basis A013609
• A003506 → x ↦ x+1 → basis A013609
• A178820 → basis A013609 → basis A013610
• A156992 → basis A007318 → basis A013609
• A038243 → basis A013611
• basis-map coefficients of A113214, shifted → D → basis A013610
• A135838 → basis A013610
• A085880 → basis A013610
• A038220 → D
• A156992 → basis A013610 → D
• A178820 → basis A013610 → D
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A178821 → basis A013610 → D
• A027465 → basis A013611 → basis A013610
• A156992 → basis A013611 → x ↦ x+1
• A038328 → basis A013609 → basis A013610
• A038334 → D
• A131431 → x ↦ x+1 → basis A013610
• A038208 → D → basis A013610
• A178822 → x ↦ x+1 → basis A013609
• A003506 → D → basis A013610
• A356546 → x ↦ x+1 → basis A013609
• A038221 → basis A013610 → D
• A085881 → basis A013609 → basis A013610
• A038325 → basis A013609 → basis A013610
• A156992 → basis A013609 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A085880 → basis A013610 → reverse
• A038338 → basis A013610 → D
• A178820 → basis A013611 → basis A007318
• A085881 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038221 → basis A013610
• A038221 → basis A013611 → x ↦ x+1
• A136572 → basis A013610 → basis A013609
• A120910 → basis A013609 → reverse
• A028326 → basis A013610 → D
• A120910 → basis A013610 → basis A013610
• A085880 → basis A013611 → x ↦ x+1
• A038208 → basis A013610
• A038325 → basis A007318 → basis A013609
• A178821 → basis A013610 → reverse
• A196347 → basis A013610 → D
• A038208 → basis A013610 → D
• A085880 → basis A013610 → D
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A134058 → basis A013610
• A094305 → x ↦ x+1 → basis A013609
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A196347 → basis A013610
• A356546 → basis A013610
• A136572 → basis A013609 → x ↦ x+1
• A038291 → basis A013609 → basis A013611
• A027466 → basis A013611 → x ↦ x/(x+1)
• A038338 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A178820 → basis A007318 → basis A013609
• A028326 → basis A013609 → basis A013610
• A216973 → basis A013610
• A135065 → basis A013610
• A134346 → basis A013610
• A356546 → basis A013609 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A013620 → D
• A178822 → basis A013609 → basis A013610
• A013610 → basis A013611 → basis A013609
• A085880 → D → basis A013610
• A085880 → basis A013609 → 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.