{"id":"n-6144534dca603004","revision":0,"first_index":0,"fingerprint":"f6-af475825caef4364","name":"one_plus_derivative of veronese_section_2_0 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],[],["4","4"],["12","12"],["44","76","16"],["200","520","160"],["1096","3816","1552","64"],["7056","30576","15792","1344"],["52272","268880","170128","21632","256"],["438336","2591424","1947456","327168","9216"],["4106304","27264064","23727296","4940352","227840","1024"],["42514560","311618560","307746560","76317120","4928000","56320"],["482175360","3850433792","4245319936","1219731392","101356288","1995776","4096"],["5945771520","51194247936","62189707008","20281362816","2058760704","59025408","319488"],["79211036160","729251135232","965554856448","351788950656","42118638848","1601553408","15396864","16384"]],"degrees":[0,-1,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","0","8","24","136","880","6528","54768","513168","5313600","60266880","743181120","9901016640","141729194880","2169541584000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0"],["4","4"],["12","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-256","128","-16"],["256","-64","0"],["-96","8","0"],["16","0","0"],["-1","0","0"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-175","56"],["148","-16"],["-42","0"],["4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-55","8"],["36","0"],["-6","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-10"],["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178156,"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-3e6a5d17ef328af2"]},{"id":178157,"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-3e6a5d17ef328af2"]},{"id":178158,"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-3e6a5d17ef328af2"]}],"region_exact":{},"incoming_edges":[{"id":"e-3e6a5d17ef328af2","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-6144534dca603004","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":"6cd869e8307506291b2aa98bb12262524fb0685985680acaf8b1497fb3e11033","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-afbf4664f43bcfae","role":"sequence","name":"veronese_section_2_0 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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:      1;\n  1:      0;\n  2:      4,       4;\n  3:     12,      12;\n  4:     44,      76,      16;\n  5:    200,     520,     160;\n  6:   1096,    3816,    1552,     64;\n  7:   7056,   30576,   15792,   1344;\n  8:  52272,  268880,  170128,  21632,  256;\n  9: 438336, 2591424, 1947456, 327168, 9216;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256-(8*n^2-64*n+128)*x+16*x^2)*P(n-4,x) + (4*n^3-42*n^2+148*n-175-(16*n-56)*x)*P(n-3,x) - (6*n^2-36*n+55-8*x)*P(n-2,x) + (4*n-10)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 0, P(2,x) = 4 + 4*x, P(3,x) = 12 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 0;\nP[2, x_] := 4 + 4*x;\nP[3, x_] := 12 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (8*n^2 - 64*n + 128)*x + 16*x^2)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 - (16*n - 56)*x)*P[n - 3, x] - (6*n^2 - 36*n + 55 - 8*x)*P[n - 2, x] + (4*n - 10)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 0; v[3] = 4 + 4*x; v[4] = 12 + 12*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (8*n^2 - 64*n + 128)*x + 16*x^2)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 - (16*n - 56)*x)*v[i - 3] - (6*n^2 - 36*n + 55 - 8*x)*v[i - 2] + (4*n - 10)*v[i - 1]); 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-60c7e36cbe887b7e","steps":["Veronese 2","D+1"],"edges":["e-5025a9cd0bd92619","e-3e6a5d17ef328af2"],"text":"A137320 \u2192 Veronese 2 \u2192 D+1"},"headline":"A137320 \u2192 Veronese 2 \u2192 D+1","chain":null}