A038281 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:         64,          176,          121;
  1:        512,         2112,         2904,          1331;
  2:       4096,        22528,        46464,         42592,          14641;
  3:      32768,       225280,       619520,        851840,         585640,         161051;
  4:     262144,      2162688,      7434240,      13629440,       14055360,        7730448,        1771561;
  5:    2097152,     20185088,     83263488,     190812160,      262366720,      216452544,       99207416,       19487171;
  6:   16777216,    184549376,    888143872,    2442395648,     4197867520,     4617654272,     3174637312,     1247178944,      214358881;
  7:  134217728,   1660944384,   9135194112,   29308747776,    60449292288,    83117776896,    76191295488,    44898441984,    15433839432,    2357947691;
  8: 1073741824,  14763950080,  91351941120,  334957117440,   805990563840,  1329884430336,  1523825909760,  1197291786240,   617353577280,  188635815280,   25937424601;
  9: 8589934592, 129922760704, 893218979840, 3684528291840, 10132452802560, 19504971644928, 26819336011776, 26340419297280, 18109038266880, 8299975872320, 2282493364888, 285311670611;

The row polynomials satisfy (1+x)*P(n,x) = (8+19*x+11*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 176*x + 121*x^2.

P[0, x_] := 64 + 176*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + 19*x + 11*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] = 64 + 176*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + 19*x + 11*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 whose (i,j)-th entry is binomial(i,j)*8^(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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • 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
        • 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
    • ↑ Iterated real-rootedness preservers (above)
    • ↑ Parent-row provers of polytool (above)

Related ?

Candidate equalities ?

• A038226 → basis A007318 → reverse 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A038301 → basis A013611 → basis A013609
• A038226 → basis A007318 → D
• A038322 → reverse
• A038328 → basis A013611 → basis A013609
• A038322: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
• A038226 → basis A007318
• A027466 → basis A013610 → x ↦ x/(x+1)
• A038322 → D

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.