A034870 → basis A013611 → basis A013610

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Submission text ?

Triangle begins:
  0:               1;
  1:              36,              144,               144;
  2:            1296,            10368,             31104,              41472,               20736;
  3:           46656,           559872,           2799360,            7464960,            11197440,              8957952,               2985984;
  4:         1679616,         26873856,         188116992,          752467968,          1881169920,           3009871872,            3009871872,            1719926784,              429981696;
  5:        60466176,       1209323520,       10883911680,        58047528960,        203166351360,         487599243264,          812665405440,          928760463360,           696570347520,           309586821120,            61917364224;
  6:      2176782336,      52242776064,      574670536704,      3831136911360,      17240116101120,       55168371523584,       128726200221696,       220673486094336,        275841857617920,        245192762327040,        147115657396224,         53496602689536,          8916100448256;
  7:     78364164096,    2194196594688,    28524555730944,    228196445847552,    1255080452161536,     5020321808646144,     15060965425938432,     34425063830716416,      60243861703753728,      80325148938338304,      80325148938338304,      58418290136973312,      29209145068486656,       8987429251842048,       1283918464548864;
  8:   2821109907456,   90275517038592,  1354132755578880,  12638572385402880,   82150720505118720,   394323458424569856,   1445852680890089472,   4131007659685969920,    9294767234293432320,   16524030638743879680,   23133642894241431552,   25236701339172470784,   21030584449310392320,   12941898122652549120,    5546527766851092480,    1479074071160291328,     184884258895036416;
  9: 101559956668416, 3656158440062976, 62154693481070592, 662983397131419648, 4972375478485647360, 27845302679519625216, 120662978277918375936, 413701639810005860352, 1137679509477516115968, 2528176687727813591040, 4550718037910064463872, 6619226236960093765632, 7722430609786776059904, 7128397485957024055296, 5091712489969302896640, 2715579994650294878208, 1018342497993860579328, 239609999527967195136, 26623333280885243904;

The row polynomials satisfy P(n,x) = (36+144*x+144*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 + 144*x + 144*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 + 144*x + 144*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 ?

• basis-map coefficients of A297191, shifted → (θ+1)² → basis A013609
• basis-map coefficients of A297191, shifted → basis A013609 → basis A013610
• A139548 → x ↦ x+1
• A139548 → x ↦ x+1 → reverse
• basis-map coefficients of A297191, shifted → basis A013609
• A034870 → x ↦ x/(x+1)
• A034870 → basis A013611
• A139548 → basis A013611
• basis-map coefficients of A297191, shifted → θ+1 → basis A013609
• A034870 → x ↦ x+1
• A034870 → basis A013609 → basis A013611
• A139548 → basis A013609 → basis A007318
• A139548 → x ↦ x/(x+1)
• A034870 → basis A013609 → 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.