{"id":"n-b2e9abf9eed0ddf3","revision":0,"first_index":1,"fingerprint":"f6-ecca132484b838e2","name":"shift_x_plus_one of derivative of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["4"],["30","20"],["168","210","60"],["840","1512","840","140"],["3960","9240","7560","2520","280"],["18018","51480","55440","27720","6300","504"],["80080","270270","360360","240240","83160","13860","840"],["350064","1361360","2162160","1801800","840840","216216","27720","1320"],["1511640","6651216","12252240","12252240","7207200","2522520","504504","51480","1980"],["6466460","31744440","66512160","77597520","55135080","24504480","6726720","1081080","90090","2860"],["27457584","148728580","349188840","465585120","387987600","209513304","73513440","16336320","2162160","150150","4004"],["115892400","686439600","1784742960","2677114440","2560718160","1629547920","698377680","199536480","36756720","4084080","240240","5460"],["486748080","3129094800","8923714800","14872858000","16062686640","11779303536","5975009040","2095133040","498841200","77597520","7351344","371280","7280"],["2035917450","14115694320","43807327200","80313433200","96673577000","80313433200","47117214144","19632172560","5761615860","1163962800","155195040","12697776","556920","9520"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","50","438","3332","23560","159462","1048810","6761480","42955020","269860890","1680627102","10393456140","63908716560","391102806990"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-2"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","2"],["-2","-3"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177605,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-2d5771442292cc52"]},{"id":177606,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-2d5771442292cc52","e-c021c4900f624753"]},{"id":177607,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-2d5771442292cc52","e-c021c4900f624753"]},{"id":177608,"position":3,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-c021c4900f624753"]}],"region_exact":{},"incoming_edges":[{"id":"e-2d5771442292cc52","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b2e9abf9eed0ddf3","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"7b50ec5e3afec563b97823c2ccbf403cb1757270018ffde475b6ec2f3582902d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-b1a4ceaaf7992b60","role":"sequence","name":"derivative of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-c021c4900f624753","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-b2e9abf9eed0ddf3","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":"33c2bf445410a5a806db51d7c586d6980cca6bbe8dc1e42ebf0625edb257b671","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-141ab74a8d7e58b7","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       0;\n   2:       4;\n   3:      30,      20;\n   4:     168,     210,       60;\n   5:     840,    1512,      840,      140;\n   6:    3960,    9240,     7560,     2520,     280;\n   7:   18018,   51480,    55440,    27720,    6300,     504;\n   8:   80080,  270270,   360360,   240240,   83160,   13860,    840;\n   9:  350064, 1361360,  2162160,  1801800,  840840,  216216,  27720,  1320;\n  10: 1511640, 6651216, 12252240, 12252240, 7207200, 2522520, 504504, 51480, 1980;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-2)*P(n,x) = (4*n^2-2*n+(3*n^2-3*n+2)*x)*P(n-1,x) - (2*n*x+2*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (3*n^2 - 3*n + 2)*x)*P[n - 1, x] - (2*n*x + 2*n*x^2)*D[P[n - 1, x], x])/(n^2 - n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 2*n + (3*n^2 - 3*n + 2)*x)*v[i - 1] - (2*n*x + 2*n*x^2)*deriv(v[i - 1], x))/(n^2 - n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a2304b34e998691c","steps":["D","x \u21a6 x+1"],"edges":["e-36b6691ae4af4747","e-2d5771442292cc52"],"text":"A062145 \u2192 D \u2192 x \u21a6 x+1"},"headline":"A062145 \u2192 D \u2192 x \u21a6 x+1","chain":null}