A023531 → basis A049020 → reverse

OEIS matches ?

• A049020: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A001861: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1,  1;
  2: 1,  3,   2;
  3: 1,  6,  10,    5;
  4: 1, 10,  31,   37,    15;
  5: 1, 15,  75,  160,   151,     52;
  6: 1, 21, 155,  520,   856,    674,    203;
  7: 1, 28, 287, 1400,  3556,   4802,   3263,    877;
  8: 1, 36, 490, 3290, 11991,  24626,  28337,  17007,  4140;
  9: 1, 45, 786, 6972, 34671, 101031, 174805, 175896, 94828, 21147;

The row polynomials satisfy P(n,x) = (1+n*x+(n-1)*x^2)*P(n-1,x) - (x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n*x + (n - 1)*x^2)*P[n - 1, x] - (x^2 + x^3)*D[P[n - 1, x], 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] = (1 + n*x + (n - 1)*x^2)*v[i - 1] - (x^2 + x^3)*deriv(v[i - 1], 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)

Triangle begins:
  0:     1;
  1:     1,     1;
  2:     2,     3,      1;
  3:     5,    10,      6,      1;
  4:    15,    37,     31,     10,      1;
  5:    52,   151,    160,     75,     15,     1;
  6:   203,   674,    856,    520,    155,    21,    1;
  7:   877,  3263,   4802,   3556,   1400,   287,   28,   1;
  8:  4140, 17007,  28337,  24626,  11991,  3290,  490,  36,  1;
  9: 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+n*x+(n-1)*x^2)*P(n-1,x) - (x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n*x + (n - 1)*x^2)*P[n - 1, x] - (x^2 + x^3)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n*x + (n - 1)*x^2)*v[i - 1] - (x^2 + x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-2a8c271a42a52261
• 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-2a8c271a42a52261
• 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-2a8c271a42a52261

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

• preserver_image reversal, shift 0 e-2a8c271a42a52261

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A049020 e-2ff0affdc4d5d50d

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

      • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d

          Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] 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)
        • parent_provers periodic_product e-b7eca039219a7f53 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-7e44b66e04c497d6

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
            • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    • preserver_image affine, shift 0 e-bcbfbcbb753a93dd

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • basis_map A008277 e-857a86cbb6c7881e

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

          • premise (sequence): basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
            • preserver_image affine, shift 0 e-0ba23af98afc4615

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
            • preserver_image affine, shift 0 e-8c1f314a0c6d10ae

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
        • preserver_image reversal, shift 1 e-ad13a5d17ca116d8

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-2ab1dfe057e84c9d root: theorem check

              Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (1)

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

• A049020: Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blo…

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
4. [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
5. [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