{"id":"n-2bfe40bbed4239a0","revision":0,"first_index":1,"fingerprint":"f6-f0aff1f976ed95f5","name":"shift_x_plus_one of derivative of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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":[[],["1"],["7","2"],["31","20","3"],["115","120","39","4"],["390","570","294","64","5"],["1254","2370","1698","580","95","6"],["3893","9048","8355","3940","1005","132","7"],["11789","32560","36939","22380","7845","1596","175","8"],["35045","112210","151302","112452","50505","14070","2380","224","9"],["102695","374160","585282","516712","284235","101220","23380","3384","279","10"],["297516","1215660","2165784","2218060","1448090","630672","185850","36648","4635","340","11"],["853932","3867852","7735560","9025180","6832610","3528504","1269954","318960","54855","6160","407","12"],["2432041","12095760","26842674","35171560","30331215","18153972","7740600","2371608","518760","79090","7986","480","13"],["6881395","37283510","90936102","132279732","128125485","87330870","43131480","15642264","4169484","807510","110550","10140","559","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","9","54","278","1323","6003","26380","113292","478197","1991357","8203266","33493986","135745759","546709095"],"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"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["-7"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-2","-1"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-6","-1"],["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174223,"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-1de068c631429cb5"]},{"id":174224,"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-1de068c631429cb5","e-c81c24435b8143b2"]},{"id":174225,"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-1de068c631429cb5","e-c81c24435b8143b2"]},{"id":174226,"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-c81c24435b8143b2"]},{"id":174227,"position":4,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-c81c24435b8143b2"]}],"region_exact":{},"incoming_edges":[{"id":"e-1de068c631429cb5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2bfe40bbed4239a0","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":"792c3c7b7ddeeabdd469209b46f861090af7d022f3dd834697704fd7dea160c2","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-a4555bce014ca1b5","role":"sequence","name":"derivative of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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-c81c24435b8143b2","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-2bfe40bbed4239a0","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":"c4e3802c5879ddb1049ffc9745e0bc1c11fe6395cf3ec46fa3b650b2811ed9dc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-67d789c805b2e2cd","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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:     1;\n   3:     7,      2;\n   4:    31,     20,      3;\n   5:   115,    120,     39,      4;\n   6:   390,    570,    294,     64,     5;\n   7:  1254,   2370,   1698,    580,    95,     6;\n   8:  3893,   9048,   8355,   3940,  1005,   132,    7;\n   9: 11789,  32560,  36939,  22380,  7845,  1596,  175,   8;\n  10: 35045, 112210, 151302, 112452, 50505, 14070, 2380, 224, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (3*n^2-2*n+(n^2-n)*x)*P(n-1,x) + (5*n-5+(6*n-6)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 2*n + (n^2 - n)*x)*P[n - 1, x] + (5*n - 5 + (6*n - 6)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((3*n^2 - 2*n + (n^2 - n)*x)*v[i - 1] + (5*n - 5 + (6*n - 6)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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-318700a428d7b97b","steps":["D","x \u21a6 x+1"],"edges":["e-05948cabe35ede38","e-1de068c631429cb5"],"text":"A111125 \u2192 D \u2192 x \u21a6 x+1"},"headline":"A111125 \u2192 D \u2192 x \u21a6 x+1","chain":null}