{"id":"n-2b7304c3ddb35f3e","revision":0,"first_index":0,"fingerprint":"f6-d97455e8c3b8e2bb","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["15","2"],["221","60","4"],["3211","1326","180","8"],["46137","25688","5304","480","16"],["656903","461370","128440","17680","1200","32"],["9282325","7882836","2768220","513760","53040","2880","64"],["130323843","129952550","55179852","12918360","1798160","148512","6720","128"],["1819706993","2085181488","1039620400","294292544","51673440","5754112","396032","15360","256"],["25287652351","32754725874","18766633392","6237722400","1324316448","186024384","17262336","1018368","34560","512"],["349948479309","505753047020","327547258740","125110889280","31188612000","5297265792","620081280","49320960","2545920","76800","1024"],["4825047214715","7698866544798","5563283517220","2402013230760","688109891040","137229892800","19423307904","1948826880","135632640","6223360","168960","2048"],["66309934579369","115801133153160","92386398537576","44506268137760","14412079384560","3302927476992","548919571200","66594198528","5846480640","361687040","14936064","368640","4096"],["908625319776759","1724058299063594","1505414730991080","800682120658992","289290742895440","74942812799712","14312685733632","2038844121600","216431145216","16889832960","940386304","35303424","798720","8192"],["12417879370282373","25441508953749252","24136816186890316","14050537489250080","5604774844612944","1620028160214464","349733126398656","57250742934528","7135954425600","673341340672","47291532288","2393710592","82374656","1720320","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","17","285","4725","77625","1265625","20503125","330328125","5296640625","84575390625","1345517578125","21336064453125","337340478515625","5319599853515625","83686387939453125"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["11","2"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["169","52","4"],["26","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188819,"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-ce85297b856b149c"]},{"id":188820,"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-ce85297b856b149c"]},{"id":188821,"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-ce85297b856b149c"]},{"id":188822,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-ce85297b856b149c"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce85297b856b149c","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-2b7304c3ddb35f3e","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":"f013b3d6beae943c21ee21268705d019f5949dd931980ec05099eb8e8bfc21ee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e88ccd05662b8cd9","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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:          15,           2;\n  2:         221,          60,           4;\n  3:        3211,        1326,         180,          8;\n  4:       46137,       25688,        5304,        480,         16;\n  5:      656903,      461370,      128440,      17680,       1200,        32;\n  6:     9282325,     7882836,     2768220,     513760,      53040,      2880,       64;\n  7:   130323843,   129952550,    55179852,   12918360,    1798160,    148512,     6720,     128;\n  8:  1819706993,  2085181488,  1039620400,  294292544,   51673440,   5754112,   396032,   15360,   256;\n  9: 25287652351, 32754725874, 18766633392, 6237722400, 1324316448, 186024384, 17262336, 1018368, 34560, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n+11+2*x)*P(n,x) = (26*n+169+(4*n+52)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((26*n + 169 + (4*n + 52)*x + 4*x^2)*P[n - 1, x])/(2*n + 11 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((26*n + 169 + (4*n + 52)*x + 4*x^2)*v[i - 1])/(2*n + 11 + 2*x)); 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-81e6fb4f9f226d77","steps":["basis A013609","D+1"],"edges":["e-ec56a67a9fb88114","e-ce85297b856b149c"],"text":"A038327 \u2192 basis A013609 \u2192 D+1"},"headline":"A038327 \u2192 basis A013609 \u2192 D+1","chain":null}