A034870 → basis A013611 → basis A013609

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Submission text ?

Triangle begins:
  0:               1;
  1:              36,               96,                64;
  2:            1296,             6912,             13824,              12288,               4096;
  3:           46656,           373248,           1244160,            2211840,            2211840,             1179648,               262144;
  4:         1679616,         17915904,          83607552,          222953472,          371589120,           396361728,            264241152,            100663296,             16777216;
  5:        60466176,        806215680,        4837294080,        17199267840,        40131624960,         64210599936,          71345111040,          54358179840,          27179089920,           8053063680,           1073741824;
  6:      2176782336,      34828517376,      255409127424,      1135151677440,      3405455032320,       7264970735616,       11301065588736,       12915503529984,       10762919608320,        6378026434560,        2551210573824,         618475290624,          68719476736;
  7:     78364164096,    1462797729792,    12677580324864,     67613761732608,    247917126352896,     661112336941056,     1322224673882112,     2014818550677504,     2350621642457088,     2089441459961856,     1392960973307904,      675375017361408,      225125005787136,       46179488366592,        4398046511104;
  8:   2821109907456,   60183678025728,   601836780257280,   3744762188267520,  16227302815825920,   51927369010642944,   126933568692682752,   241778226081300480,   362667339121950720,   429827957477867520,   401172760312676352,   291762007500128256,   162090004166737920,    66498463247892480,    18999560927969280,    3377699720527872,     281474976710656;
  9: 101559956668416, 2437438960041984, 27624308213809152, 196439525075976192, 982197625379880960, 3666871134751555584, 10593183278171160576, 24212990350105509888, 44390482308526768128, 65763677494113730560, 78916412992936476672, 76525006538605068288, 59519449530026164224, 36627353556939177984, 17441596931875799040, 6201456686889172992, 1550364171722293248, 243194379878006784, 18014398509481984;

The row polynomials satisfy P(n,x) = (36+96*x+64*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(36 + 96*x + 64*x^2)*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] = (36 + 96*x + 64*x^2)*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Even-numbered rows of Pascal's triangle.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Even-numbered rows of Pascal's triangle.?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Even-numbered rows of Pascal's triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • 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
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • 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
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Even-numbered rows of Pascal's triangle.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Even-numbered rows of Pascal's triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Even-numbered rows of Pascal's triangle.?
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A034870 → x ↦ x+1 → basis A013611
• basis-map coefficients of A297191, shifted → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.