{"id":"n-9a319b53d7728bfd","revision":0,"first_index":0,"fingerprint":"f6-b118b3db7c0b274c","name":"one_plus_derivative of veronese_section_2_1 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["3"],["9","4"],["18","12"],["46","54","8"],["72","135","36"],["140","465","216","16"],["192","900","648","96"],["318","2308","2778","720","32"],["405","3756","6363","2520","240"],["607","7896","20153","12860","2160","64"],["738","11556","37926","33876","8640","576"],["1034","21270","96570","127214","51000","6048","128"],["1218","29043","159840","271407","151452","27216","1344"],["1626","48685","349176","810561","655984","182112","16128","256"],["1872","63336","529776","1497096","1562448","600768","80640","3072"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["3","13","30","108","243","837","1836","6156","13284","43740","93312","303264","641520","2064528","4339008"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"],["9","4"],["18","12"],["46","54","8"],["72","135","36"],["140","465","216","16"]],"recurrence":{"denominator":{"coeffs":[["32"]]},"terms":[{"coeff":{"coeffs":[["-625","1731","-1164","-320"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["81","-549","1044","-576"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-1050","732","192"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-162","-36","576"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["249","48"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["81","-144"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174056,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-637cd558485a27da"]},{"id":174057,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-637cd558485a27da"]},{"id":174058,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-637cd558485a27da"]}],"region_exact":{},"incoming_edges":[{"id":"e-637cd558485a27da","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9a319b53d7728bfd","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"9d5f1a6c2ea1c7d0ef89c388b3f107d09bd5e8a7ab68c4e1d8a9d8547b114d5a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cfed7b5cee98d33b","role":"sequence","name":"veronese_section_2_1 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   3;\n  1:   9,    4;\n  2:  18,   12;\n  3:  46,   54,     8;\n  4:  72,  135,    36;\n  5: 140,  465,   216,    16;\n  6: 192,  900,   648,    96;\n  7: 318, 2308,  2778,   720,   32;\n  8: 405, 3756,  6363,  2520,  240;\n  9: 607, 7896, 20153, 12860, 2160, 64;\n","recurrence_sentence":"The row polynomials satisfy 32*P(n,x) = -(625-1731*x+1164*x^2+320*x^3)*P(n-6,x) + (81-549*x+1044*x^2-576*x^3)*d/dx P(n-6,x) - (1050-732*x-192*x^2)*P(n-4,x) - (162+36*x-576*x^2)*d/dx P(n-4,x) + (249+48*x)*P(n-2,x) + (81-144*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 3, P(1,x) = 9 + 4*x, P(2,x) = 18 + 12*x, P(3,x) = 46 + 54*x + 8*x^2, P(4,x) = 72 + 135*x + 36*x^2, P(5,x) = 140 + 465*x + 216*x^2 + 16*x^3.","mathematica":"P[0, x_] := 3;\nP[1, x_] := 9 + 4*x;\nP[2, x_] := 18 + 12*x;\nP[3, x_] := 46 + 54*x + 8*x^2;\nP[4, x_] := 72 + 135*x + 36*x^2;\nP[5, x_] := 140 + 465*x + 216*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(625 - 1731*x + 1164*x^2 + 320*x^3)*P[n - 6, x] + (81 - 549*x + 1044*x^2 - 576*x^3)*D[P[n - 6, x], x] - (1050 - 732*x - 192*x^2)*P[n - 4, x] - (162 + 36*x - 576*x^2)*D[P[n - 4, x], x] + (249 + 48*x)*P[n - 2, x] + (81 - 144*x)*D[P[n - 2, x], x])/(32)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 9 + 4*x; v[3] = 18 + 12*x; v[4] = 46 + 54*x + 8*x^2; v[5] = 72 + 135*x + 36*x^2; v[6] = 140 + 465*x + 216*x^2 + 16*x^3; for(n = 6, N, my(i = n + 1); v[i] = (-(625 - 1731*x + 1164*x^2 + 320*x^3)*v[i - 6] + (81 - 549*x + 1044*x^2 - 576*x^3)*deriv(v[i - 6], x) - (1050 - 732*x - 192*x^2)*v[i - 4] - (162 + 36*x - 576*x^2)*deriv(v[i - 4], x) + (249 + 48*x)*v[i - 2] + (81 - 144*x)*deriv(v[i - 2], x))/(32)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5a70a6e97131d437","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-1f83abdcb5264ea5","e-637cd558485a27da"],"text":"A064861 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A064861 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}