A038338 → basis A013610 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:              576,              2880,               3600;
  1:            13824,            103680,             259200,              216000;
  2:           331776,           3317760,           12441600,            20736000,             12960000;
  3:          7962624,          99532800,          497664000,          1244160000,           1555200000,            777600000;
  4:        191102976,        2866544640,        17915904000,         59719680000,         111974400000,         111974400000,           46656000000;
  5:       4586471424,       80263249920,       601974374400,       2508226560000,        6270566400000,        9405849600000,         7838208000000,         2799360000000;
  6:     110075314176,     2201506283520,     19263179980800,      96315899904000,      300987187200000,      601974374400000,       752467968000000,       537477120000000,       167961600000000;
  7:    2641807540224,    59440669655040,    594406696550400,    3467372396544000,    13002646487040000,    32506616217600000,     54177693696000000,     58047528960000000,     36279705600000000,     10077696000000000;
  8:   63403380965376,  1585084524134400,  17832200896512000,  118881339310080000,   520105859481600000,  1560317578444800000,   3250661621760000000,   4643802316800000000,   4353564672000000000,   2418647040000000000,    604661760000000000;
  9: 1521681143169024, 41846231437148160, 523077892964352000, 3923084197232640000, 19615420986163200000, 68653973451571200000, 171634933628928000000, 306490952908800000000, 383113691136000000000, 319261409280000000000, 159630704640000000000, 36279705600000000000;

The row polynomials satisfy (1+x)*P(n,x) = (24+84*x+60*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 2880*x + 3600*x^2.

P[0, x_] := 576 + 2880*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24 + 84*x + 60*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 2880*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24 + 84*x + 60*x^2)*v[i - 1])/(1 + x)); 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, 0]
  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 Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

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

  • from basis_map_A013610 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 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) (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 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)

Related ?

Same fingerprint ?

• A038220 → basis A007318 → D
• A013620 → x ↦ x/(x+1)
• A178821 → basis A013610 → x ↦ x/(x+1)
• A038291 → basis A013611 → reverse
• A085880 → basis A013610 → x ↦ x/(x+1)
• A038291 → basis A013611 → D
• A038220 → basis A007318 → reverse
• A038231 → D → basis A013609
• A038334 → x ↦ x+1
• A038327 → basis A013609 → basis A013610
• A038291 → basis A013611
• A038329 → basis A013609
• A038220 → basis A007318
• A120910 → basis A013609 → basis A007318
• A156992 → basis A013610 → x ↦ x/(x+1)
• A196347 → basis A013610 → x ↦ x/(x+1)
• A038221 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A013611 → D
• A013612 → basis A013610 → D
• A013622 → basis A013611 → reverse
• A013622 → basis A013611
• A038208 → basis A013610 → x ↦ x/(x+1)
• A178820 → basis A013610 → x ↦ x/(x+1)
• A038329 → basis A013609 → D
• A178822 → basis A013610 → x ↦ x/(x+1)
• A028326 → basis A013610 → x ↦ x/(x+1)
• A013612 → basis A013610
• A038334 → x ↦ x+1 → D
• A038334 → x ↦ x+1 → reverse

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.