A038331 → reverse

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Submission text ?

Triangle begins:
  0:       25,        120,         144;
  1:      125,        900,        2160,         1728;
  2:      625,       6000,       21600,        34560,        20736;
  3:     3125,      37500,      180000,       432000,       518400,        248832;
  4:    15625,     225000,     1350000,      4320000,      7776000,       7464960,       2985984;
  5:    78125,    1312500,     9450000,     37800000,     90720000,     130636800,     104509440,      35831808;
  6:   390625,    7500000,    63000000,    302400000,    907200000,    1741824000,    2090188800,    1433272320,     429981696;
  7:  1953125,   42187500,   405000000,   2268000000,   8164800000,   19595520000,   31352832000,   32248627200,   19349176320,    5159780352;
  8:  9765625,  234375000,  2531250000,  16200000000,  68040000000,  195955200000,  391910400000,  537477120000,  483729408000,  257989017600,   61917364224;
  9: 48828125, 1289062500, 15468750000, 111375000000, 534600000000, 1796256000000, 4311014400000, 7390310400000, 8868372480000, 7094697984000, 3405455032320, 743008370688;

The row polynomials satisfy P(n,x) = (5+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 120*x + 144*x^2.

P[0, x_] := 25 + 120*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 25 + 120*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 12*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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • 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 ?

• A178822 → basis A013610 → basis A013611
• A136572 → basis A013611 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• A038271 → basis A007318
• A013611 → D → basis A013610
• A028326 → basis A013610 → basis A013611
• A013620 → basis A013611 → D
• A085880 → basis A013610 → basis A013611
• A038271 → basis A007318 → D
• A130123 → basis A013611 → basis A013610
• A178820 → basis A013610 → basis A013611
• A038331 → D
• A178821 → basis A013610 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → basis A013610
• basis-map coefficients of A112227 → basis A013611 → basis A013610
• A013620 → basis A013611

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.