A038338 → basis A013610 → D+1

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Submission text ?

Triangle begins:
  0:              2304,               4320,                1296;
  1:             76032,             248832,              233280,                46656;
  2:           2322432,           10948608,            17915904,             11197440,              1679616;
  3:          67682304,          418037760,           985374720,           1074954240,            503884800,             60466176;
  4:        1911029760,        14619377664,         45148078080,          70946979840,          58047528960,          21767823360,           2176782336;
  5:       52744421376,       481579499520,       1842041585664,        3792438558720,        4469659729920,        2925595459584,         914248581120,          78364164096;
  6:     1430979084288,     15190393356288,      69347447930880,      176835992223744,      273055576227840,      257452400443392,      140428582060032,       37614798766080,        2821109907456;
  7:    38306209333248,    463637223309312,    2460843723718656,     7489524376535040,    14323715370123264,    17694001339564032,    13902429623943168,     6499837226778624,     1523399350026240,      101559956668416;
  8:  1014454095446016,  13790235359969280,   83454700195676160,   295301246846238720,   674057193888153600,  1031307506648875008,  1061640080373841920,   714982094945648640,   292492675205038080,    60935974001049600,    3656158440062976;
  9: 26629420005457920, 401723821796622336, 2730466601273917440, 11016020425829253120, 29234823437777633280, 53385329755941765120, 68066295438825750528, 60058495975434485760, 35391613699809607680, 12869677709021675520, 2413064570441564160, 131621703842267136;

The row polynomials satisfy (3*n+5+3*x)*P(n,x) = (72*n+192+(108*n+360)*x+108*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2304 + 4320*x + 1296*x^2.

P[0, x_] := 2304 + 4320*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72*n + 192 + (108*n + 360)*x + 108*x^2)*P[n - 1, x])/(3*n + 5 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2304 + 4320*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72*n + 192 + (108*n + 360)*x + 108*x^2)*v[i - 1])/(3*n + 5 + 3*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 (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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
        • 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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A038221 → basis A013610 → D+1
• A178820 → basis A013610 → D+1
• A013620 → D+1
• A038208 → basis A013610 → D+1
• A196347 → basis A013610 → D+1
• A028326 → basis A013610 → D+1
• A178822 → basis A013610 → D+1
• A178821 → basis A013610 → D+1
• A085880 → basis A013610 → D+1
• A156992 → basis A013610 → D+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.