A034870 → basis A008277

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OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:      1;
  1:      4,         5,           1;
  2:     16,        65,          55,           14,             1;
  3:     64,       665,        1351,          910,           245,            27,              1;
  4:    256,      6305,       26335,        35574,         20181,          5418,            714,             44,             1;
  5:   1024,     58025,      465751,      1132670,       1144165,        563409,         147147,          21120,          1650,            65,            1;
  6:   4096,    527345,     7859215,     32566534,      54115061,      43613856,       19012708,        4775628,        713427,         63635,         3289,          90,          1;
  7:  16384,   4766585,   129442951,    885423630,    2314233285,    2873141271,     1925136213,      753655760,     181092340,      27457430,      2650648,      161070,       5915,       119,        1;
  8:  65536,  42981185,  2104469695,  23305343894,   93181501141,  171754378614,   168620069982,    96646573452,   34353829653,    7878943930,   1194306542,   120944460,    8158878,    359380,     9860,    152,     1;
  9: 262144, 387158345, 33972448951, 601616805790, 3612997293605, 9650629410813, 13461181659199, 10866668017920, 5440287930870, 1771429211695, 387549682091, 58176221220, 6058947050, 438412422, 21823818, 728688, 15504, 189, 1;

The row polynomials satisfy P(n,x) = (4+5*x+x^2)*P(n-1,x) + (5*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + 5*x + x^2)*P[n - 1, x] + (5*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
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] = (4 + 5*x + x^2)*v[i - 1] + (5*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-d4da514254176df5
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-d4da514254176df5
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-d4da514254176df5

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A008277 e-d4da514254176df5

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Even-numbered rows of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-0029b53144d984cf root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-a3e5463f21dc8d9d

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Even-numbered rows of Pascal's triangle.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-0029b53144d984cf (expanded above)
        • basis_map A007318 e-fc612a08cfe0bbb6

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
        • preserver_image affine, shift 0 e-5b79b5b4d4f1c577

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-12303ad0e0b9f8eb (expanded above)
    • ↵ basis_map A007318 e-fc612a08cfe0bbb6 (expanded above)
    • ↵ preserver_image affine, shift 0 e-5b79b5b4d4f1c577 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A139548 → basis A008277

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741