A038338 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:              1728,               2592;
  1:             62208,             186624,              139968;
  2:           1990656,            8957952,            13436928,              6718464;
  3:          59719680,          358318080,           806215680,            806215680,            302330880;
  4:        1719926784,        12899450880,         38698352640,          58047528960,          43535646720,          13060694016;
  5:       48157949952,       433421549568,       1625330810880,        3250661621760,        3656994324480,        2194196594688,         548549148672;
  6:     1320903770112,     13869489586176,      62412703137792,      156031757844480,      234047636766720,      210642873090048,      105321436545024,       22568879259648;
  7:    35664401793024,    427972821516288,    2246857312960512,     6740571938881536,    12638572385402880,    15166286862483456,    11374715146862592,     4874877920083968,      914039610015744;
  8:   951050714480640,  12839184645488640,   77035107872931840,   269622877555261440,   606651474499338240,   909977211749007360,   909977211749007360,   584985350410076160,   219369506403778560,   36561584400629760;
  9: 25107738862288896, 376616082934333440, 2542158559806750720, 10168634239227002880, 26692664877970882560, 48046796780347588608, 60058495975434485760, 51478710836086702080, 28956774845298769920, 9652258281766256640, 1447838742264938496;

The row polynomials satisfy (n+1)*P(n,x) = (24*n+48+(36*n+72)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1728 + 2592*x.

P[0, x_] := 1728 + 2592*x;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 48 + (36*n + 72)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1728 + 2592*x; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 48 + (36*n + 72)*x)*v[i - 1])/(n + 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 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
• A085881 → basis A013609 → basis A013610
• A094305 → basis A013610
• A156992 → basis A007318 → basis A013609
• A085880 → basis A013609 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A085881 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A038221 → basis A013609 → basis A013610
• A038208 → basis A013610
• A027466 → basis A013611 → x ↦ x/(x+1)
• A085880 → basis A013610
• A003506 → D → basis A013610
• A178821 → basis A013610 → reverse
• A085881 → basis A013611 → basis A007318
• A135089 → basis A013610
• A178820 → basis A013610 → D
• A085880 → basis A013610 → reverse
• A136572 → basis A013609 → x ↦ x+1
• A085880 → basis A013610 → D
• A038334 → reverse
• A344565 → basis A013610
• A094305 → x ↦ x+1 → basis A013609
• A156992 → basis A013610
• A178820 → basis A013609 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• A038334 → D
• A258758 → basis A013610
• A178821 → basis A013611 → basis A007318
• A135065 → x ↦ x+1 → basis A013609
• A003506 → x ↦ x+1 → basis A013609
• A038325 → basis A013610
• A038338 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A027465 → basis A013611 → basis A013610
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A178820 → basis A007318 → basis A013609
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A038243 → basis A013611 → D
• A038243 → basis A013611
• A124860 → basis A013610
• A356546 → basis A013610
• A038328 → basis A013609 → basis A013610
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A135838 → basis A013610
• A028326 → basis A013610
• A028326 → basis A013610 → D
• A136572 → basis A013610 → basis A013609
• basis-map coefficients of A113214, shifted → basis A013610
• basis-map coefficients of A113214, shifted → D → basis A013610
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A038208 → basis A013610 → D
• A376827 → basis A013610
• A085880 → basis A013611 → x ↦ x+1
• A178821 → basis A013610
• A038325 → basis A007318 → basis A013609
• A196347 → basis A013611 → x ↦ x+1
• A356546 → basis A013611 → basis A007318
• A013609 → basis A013610 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A135065 → basis A013610
• A178820 → basis A013610
• A134346 → basis A013610
• A156992 → basis A013609 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A156992 → basis A013610 → D
• A356546 → basis A013609 → basis A013610
• A132200 → basis A013610
• A038330 → basis A013611 → basis A013610
• A085881 → basis A007318 → basis A013609
• A038220 → D
• A120910 → basis A013610 → basis A013610
• A120910 → basis A013609
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038330 → basis A013609 → x ↦ x/(x+1)
• A178820 → basis A013611 → basis A007318
• A178822 → basis A013610 → D
• A028326 → basis A013611 → x ↦ x+1
• A178822 → basis A013611 → basis A007318
• A356546 → x ↦ x+1 → basis A013609
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A120910 → basis A013609 → reverse
• A013610 → D → basis A013609
• A028326 → basis A007318 → basis A013609
• A178821 → basis A013610 → D
• A013610 → basis A013611 → basis A013609
• A131431 → x ↦ x+1 → basis A013610
• A120910 → D → basis A013609
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A178822 → x ↦ x+1 → basis A013609
• A038221 → basis A013610 → D
• A038291 → basis A013609 → basis A013611
• A178821 → basis A013609 → basis A013610
• A028326 → D → basis A013610
• A134058 → basis A013610
• A178822 → basis A013609 → basis A013610
• A196347 → basis A013610
• A038221 → basis A013611 → x ↦ x+1
• A178822 → basis A013610
• A085880 → D → basis A013610
• A134059 → basis A013610
• A038338 → basis A013610 → reverse
• A216973 → basis A013610
• A120910 → basis A013609 → D
• A196347 → basis A013610 → D
• A156992 → basis A013611 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• A028326 → basis A013609 → basis A013610
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A038330 → basis A013609 → basis A013609
• A013620 → D
• A038338 → basis A013609 → basis A013610
• basis-map coefficients of A127647 → basis A013610
• A038325 → basis A013609 → basis A013610
• A120910 → x ↦ x/(x+1)
• A038208 → D → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.

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.