A038305 → reverse

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Submission text ?

Triangle begins:
  0:      9,      60,       100;
  1:     27,     270,       900,       1000;
  2:     81,    1080,      5400,      12000,      10000;
  3:    243,    4050,     27000,      90000,     150000,      100000;
  4:    729,   14580,    121500,     540000,    1350000,     1800000,      1000000;
  5:   2187,   51030,    510300,    2835000,    9450000,    18900000,     21000000,     10000000;
  6:   6561,  174960,   2041200,   13608000,   56700000,   151200000,    252000000,    240000000,    100000000;
  7:  19683,  590490,   7873200,   61236000,  306180000,  1020600000,   2268000000,   3240000000,   2700000000,   1000000000;
  8:  59049, 1968300,  29524500,  262440000, 1530900000,  6123600000,  17010000000,  32400000000,  40500000000,  30000000000,  10000000000;
  9: 177147, 6495390, 108256500, 1082565000, 7217100000, 33679800000, 112266000000, 267300000000, 445500000000, 495000000000, 330000000000, 100000000000;

The row polynomials satisfy P(n,x) = (3+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 60*x + 100*x^2.

P[0, x_] := 9 + 60*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 9 + 60*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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 (−∞, 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 Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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

Related ?

Same fingerprint ?

• A038335 → x ↦ x+1 → basis A007318
• A013612 → basis A013611 → D
• A038305 → D
• A013612 → basis A013611
• A038291 → basis A013610
• A038291 → basis A013610 → reverse
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• A038291 → basis A013610 → D
• A038284 → basis A007318 → x ↦ x+1

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.