A013612 → basis A013609 → D+1

Matches ?

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Submission text ?

Triangle begins:
  0:        156,          320,          100;
  1:       1296,         4680,         4800,          1000;
  2:       9936,        51840,        93600,         64000,          10000;
  3:      72576,       496800,      1296000,       1560000,         800000,         100000;
  4:     513216,      4354560,     14904000,      25920000,       23400000,        9600000,        1000000;
  5:    3545856,     35925120,    152409600,     347760000,      453600000,      327600000,      112000000,       10000000;
  6:   24074496,    283668480,   1437004800,    4064256000,     6955200000,     7257600000,     4368000000,     1280000000,      100000000;
  7:  161243136,   2166704640,  12765081600,   43110144000,    91445760000,   125193600000,   108864000000,    56160000000,    14400000000,    1000000000;
  8: 1068235776,  16124313600, 108335232000,  425502720000,  1077753600000,  1828915200000,  2086560000000,  1555200000000,   702000000000,  160000000000,   10000000000;
  9: 7014076416, 117505935360, 886837248000, 3972291840000, 11701324800000, 23710579200000, 33530112000000, 32788800000000, 21384000000000, 8580000000000, 1760000000000, 100000000000;

The row polynomials satisfy (5*n+8+5*x)*P(n,x) = (30*n+78+(50*n+160)*x+50*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 156 + 320*x + 100*x^2.

P[0, x_] := 156 + 320*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 78 + (50*n + 160)*x + 50*x^2)*P[n - 1, x])/(5*n + 8 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 156 + 320*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((30*n + 78 + (50*n + 160)*x + 50*x^2)*v[i - 1])/(5*n + 8 + 5*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_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (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 ?

• A013622 → D+1
• A038271 → basis A013611 → 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.