A013612 → basis A013609 → reverse

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Submission text ?

Triangle begins:
  0:          100,          120,            36;
  1:         1000,         1800,          1080,           216;
  2:        10000,        24000,         21600,          8640,          1296;
  3:       100000,       300000,        360000,        216000,         64800,          7776;
  4:      1000000,      3600000,       5400000,       4320000,       1944000,        466560,         46656;
  5:     10000000,     42000000,      75600000,      75600000,      45360000,      16329600,       3265920,       279936;
  6:    100000000,    480000000,    1008000000,    1209600000,     907200000,     435456000,     130636800,     22394880,      1679616;
  7:   1000000000,   5400000000,   12960000000,   18144000000,   16329600000,    9797760000,    3919104000,   1007769600,    151165440,    10077696;
  8:  10000000000,  60000000000,  162000000000,  259200000000,  272160000000,  195955200000,   97977600000,  33592320000,   7558272000,  1007769600,   60466176;
  9: 100000000000, 660000000000, 1980000000000, 3564000000000, 4276800000000, 3592512000000, 2155507200000, 923788800000, 277136640000, 55427328000, 6651279360, 362797056;

The row polynomials satisfy P(n,x) = (10+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 120*x + 36*x^2.

P[0, x_] := 100 + 120*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(10 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 100 + 120*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + 6*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 basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • 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

Related ?

Same fingerprint ?

• A013620 → basis A007318 → D
• A178822 → basis A013610 → basis A007318
• A028326 → basis A013610 → x ↦ x+1
• A038319 → basis A013611 → basis A013610
• A038329 → basis A013610 → reverse
• A156992 → basis A013610 → x ↦ x+1
• A038243 → basis A013611 → x ↦ x/(x+1)
• A038284 → basis A007318 → basis A013609
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
• A038329 → basis A013610 → D
• A013612 → basis A013609
• A038221 → basis A013610 → basis A007318
• A038255 → basis A013610 → basis A013609
• A013622 → D
• A196347 → basis A013610 → basis A007318
• A013620 → basis A007318
• A038271 → basis A013610 → basis A013610
• A013612 → basis A013609 → D
• A038329 → basis A013610
• A038271 → basis A013611 → D
• A038271 → basis A013611
• A038231 → D → basis A013610
• A038335 → x ↦ x+1 → basis A013609
• A085880 → basis A013610 → x ↦ x+1
• A178821 → basis A013610 → x ↦ x+1
• A120910 → basis A013609 → x ↦ x/(x+1)
• A038338 → basis A013610 → x ↦ x+1
• A178820 → basis A013610 → x ↦ x+1
• A027467 → basis A013611 → basis A013610
• A038334 → x ↦ x/(x+1)
• A013622 → 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.