A038330 → basis A007318 → basis A007318

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Submission text ?

Triangle begins:
  0:             400,             160,              16;
  1:            8000,            4800,             960,              64;
  2:          160000,          128000,           38400,            5120,             256;
  3:         3200000,         3200000,         1280000,          256000,           25600,           1024;
  4:        64000000,        76800000,        38400000,        10240000,         1536000,         122880,          4096;
  5:      1280000000,      1792000000,      1075200000,       358400000,        71680000,        8601600,        573440,        16384;
  6:     25600000000,     40960000000,     28672000000,     11468800000,      2867200000,      458752000,      45875200,      2621440,       65536;
  7:    512000000000,    921600000000,    737280000000,    344064000000,    103219200000,    20643840000,    2752512000,    235929600,    11796480,     262144;
  8:  10240000000000,  20480000000000,  18432000000000,   9830400000000,   3440640000000,   825753600000,  137625600000,  15728640000,  1179648000,   52428800,   1048576;
  9: 204800000000000, 450560000000000, 450560000000000, 270336000000000, 108134400000000, 30277632000000, 6055526400000, 865075200000, 86507520000, 5767168000, 230686720, 4194304;

The row polynomials satisfy P(n,x) = (20+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 160*x + 16*x^2.

P[0, x_] := 400 + 160*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 400 + 160*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 4*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)*4^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis_map_A007318 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (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)*4^j.?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• Triangle of coefficients in expansion of (1+5x)^n.
• A013612 → D
• A038243 → reverse
• A038329 → x ↦ x+1
• A013612 → reverse
• A027467 → basis A013611 → x ↦ x+1
• A038329 → x ↦ x+1 → reverse
• A147716 → basis A013610 → D
• A147716 → basis A013610 → reverse
• A013611 → x ↦ x/(x+1)
• A130123 → basis A013611 → x ↦ x/(x+1)
• A038329 → x ↦ x+1 → D
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
• basis-map coefficients of A026729 → basis A013611 → x ↦ x/(x+1)
• A038291 → basis A013609 → D
• A038291 → basis A013609 → reverse
• A147716 → basis A013610
• basis-map coefficients of A112227 → basis A013611 → x ↦ x/(x+1)
• A038243 → D
• A136572 → basis A013611 → x ↦ x/(x+1)
• A038291 → basis A013609

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.