A013612 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0: 1, 12,   36;
  1: 1, 18,  108,   216;
  2: 1, 24,  216,   864,   1296;
  3: 1, 30,  360,  2160,   6480,    7776;
  4: 1, 36,  540,  4320,  19440,   46656,    46656;
  5: 1, 42,  756,  7560,  45360,  163296,   326592,   279936;
  6: 1, 48, 1008, 12096,  90720,  435456,  1306368,  2239488,   1679616;
  7: 1, 54, 1296, 18144, 163296,  979776,  3919104, 10077696,  15116544,  10077696;
  8: 1, 60, 1620, 25920, 272160, 1959552,  9797760, 33592320,  75582720, 100776960,  60466176;
  9: 1, 66, 1980, 35640, 427680, 3592512, 21555072, 92378880, 277136640, 554273280, 665127936, 362797056;

The row polynomials satisfy (1+x)*P(n,x) = (1+7*x+6*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 36*x^2.

P[0, x_] := 1 + 12*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 7*x + 6*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] = 1 + 12*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 7*x + 6*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 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 ?

• A038255 → D
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
• A038291 → basis A013609 → basis A007318
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• A038329 → x ↦ x+1 → x ↦ x+1
• A013613 → D
• A038328 → D
• A038328 → reverse
• A013613: Triangle of coefficients in expansion of (1+6x)^n.
• A147716 → basis A013610 → 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.