A038271 → basis A013609 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:             1;
  1:            22,            10;
  2:           484,           440,           100;
  3:         10648,         14520,          6600,          1000;
  4:        234256,        425920,        290400,         88000,         10000;
  5:       5153632,      11712800,      10648000,       4840000,       1100000,        100000;
  6:     113379904,     309217920,     351384000,     212960000,      72600000,      13200000,      1000000;
  7:    2494357888,    7936593280,   10822627200,    8198960000,    3726800000,    1016400000,    154000000,     10000000;
  8:   54875873536,  199548631040,  317463731200,  288603392000,  163979200000,   59628800000,  13552000000,   1760000000,   100000000;
  9: 1207269217792, 4938828618240, 8979688396800, 9523911936000, 6493576320000, 2951625600000, 894432000000, 174240000000, 19800000000, 1000000000;

The row polynomials satisfy P(n,x) = (22+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(22 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (22 + 10*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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038301 → basis A013611
• A038336 → x ↦ x+1 → D
• A038314 → x ↦ x/(x+1)
• A038301 → basis A013611 → reverse
• A038301 → basis A013611 → D
• A038259 → basis A007318 → reverse
• A013618 → basis A013610 → basis A013610
• A038220 → basis A013610 → x ↦ x/(x+1)
• A038319 → reverse
• A038336 → x ↦ x+1
• A038259 → basis A007318
• A038336 → x ↦ x+1 → reverse
• A038319 → D
• A038259 → basis A007318 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.

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.