{"id":"n-04dc627a0f0084ad","revision":0,"first_index":0,"fingerprint":"f6-fa8b89dcb83e73ac","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["8"],["304","128"],["8664","7296","1536"],["219488","277248","116736","16384"],["5212840","8779520","5544960","1556480","163840"],["118852752","250216320","210708480","88719360","18677760","1572864"],["2634569336","6655754112","7006056960","3933224960","1242071040","209190912","14680064"],["57207791296","168612437504","212984131584","149462548480","62931599360","15898509312","2231369728","134217728"],["1222816538952","4118960973312","6070047750144","5111619158016","2690325872640","906215030784","190782111744","22951231488","1207959552"],["25815015822320","97825323116160","164758438932480","161867940003840","102232383160320","43045213962240","12082867077120","2180366991360","229512314880","10737418240"],["539533830686488","2271721392364160","4304314217111040","4832914208686080","3561094680084480","1799289943621632","631329804779520","151898900398080","23984036904960","2244120412160","94489280512"],["11183064854229024","51795247745902848","109042626833479680","137738054947553280","115989941008465920","68373017857622016","28788639097946112","8658237322690560","1822786804776960","255829726986240","21543555956736","824633720832"],["230184751582880744","1163038744839818496","2693352882786948096","3780144396893962240","3581189428636385280","2412590772976091136","1185132309532114944","427716923740913664","112557085194977280","21063314188533760","2660629160656896","203684529045504","7146825580544"],["4709934147772790608","25780692177282643328","65130169711029835776","100551840957379395584","105844043113030942720","80218643201455030272","45035027762220367872","18962116952513839104","5988036932372791296","1400710393537495040","235909118911578112","27090042363052032","1901055604424704","61572651155456"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","8","432","17496","629856","21257640","688747536","21695547384","669462604992","20334926626632","610047798798960","18118419624329112","533669814389330208","15609842070887908584","453886177138125341904"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["8"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["19","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191869,"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-154c3ca700609059"]},{"id":191870,"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-154c3ca700609059"]},{"id":191871,"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-154c3ca700609059"]},{"id":191872,"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-154c3ca700609059"]}],"region_exact":{},"incoming_edges":[{"id":"e-154c3ca700609059","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-04dc627a0f0084ad","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":"ce517b084f5442aec0b2015025b801e94fb0515bd971d8e7cd55f32836598401","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-0d855b65b3c75e99","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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":[{"id":"l-a3cbc4b222bc1224","kind":"fingerprint_related","node_a_id":"n-04dc627a0f0084ad","node_b_id":"n-36db20e01be4b1a5","fingerprint":"f6-fa8b89dcb83e73ac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fa8b89dcb83e73ac"},"other_id":"n-36db20e01be4b1a5","other":{"id":"n-36db20e01be4b1a5","name":"shift_x_plus_one of basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-fa8b89dcb83e73ac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["361","304","64"],["6859","8664","3648","512"],["130321","219488","138624","38912","4096"]],"card_rows":[["361","304","64"],["6859","8664","3648","512"],["130321","219488","138624","38912","4096"],["2476099","5212840","4389760","1848320","389120","32768"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A007318","basis A007318"],"edges":["e-0566c972c7be0ca0","e-47286d53ecaa66a6"],"text":"A038226 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A038226 \u2192 basis A007318 \u2192 basis A007318","chain":null}},{"id":"l-d02daff133208b67","kind":"fingerprint_related","node_a_id":"n-0d855b65b3c75e99","node_b_id":"n-04dc627a0f0084ad","fingerprint":"f6-fa8b89dcb83e73ac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fa8b89dcb83e73ac"},"other_id":"n-0d855b65b3c75e99","other":{"id":"n-0d855b65b3c75e99","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-fa8b89dcb83e73ac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["19","8"],["361","304","64"]],"card_rows":[["1"],["19","8"],["361","304","64"],["6859","8664","3648","512"]],"row_sums_preview":["1","27","729","19683","531441","14348907","387420489","10460353203"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["basis A007318"],"edges":["e-c22af348c5e552a2"],"text":"A038322 \u2192 basis A007318"},"headline":"A038322 \u2192 basis A007318","chain":null}},{"id":"l-d3c6827c42b9a7b2","kind":"fingerprint_related","node_a_id":"n-04dc627a0f0084ad","node_b_id":"n-7d8dce85fe57eb21","fingerprint":"f6-fa8b89dcb83e73ac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fa8b89dcb83e73ac"},"other_id":"n-7d8dce85fe57eb21","other":{"id":"n-7d8dce85fe57eb21","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-fa8b89dcb83e73ac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","19"],["64","304","361"]],"card_rows":[["1"],["8","19"],["64","304","361"],["512","3648","8664","6859"]],"row_sums_preview":["1","27","729","19683","531441","14348907","387420489","10460353203"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["basis A007318","reverse"],"edges":["e-c22af348c5e552a2","e-a48b7726e574c7df"],"text":"A038322 \u2192 basis A007318 \u2192 reverse"},"headline":"A038322 \u2192 basis A007318 \u2192 reverse","chain":null}}],"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:             0;\n  1:             8;\n  2:           304,           128;\n  3:          8664,          7296,          1536;\n  4:        219488,        277248,        116736,         16384;\n  5:       5212840,       8779520,       5544960,       1556480,        163840;\n  6:     118852752,     250216320,     210708480,      88719360,      18677760,      1572864;\n  7:    2634569336,    6655754112,    7006056960,    3933224960,    1242071040,    209190912,     14680064;\n  8:   57207791296,  168612437504,  212984131584,  149462548480,   62931599360,  15898509312,   2231369728,   134217728;\n  9: 1222816538952, 4118960973312, 6070047750144, 5111619158016, 2690325872640, 906215030784, 190782111744, 22951231488, 1207959552;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (19*n+8*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[((19*n + 8*n*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((19*n + 8*n*x)*v[i - 1])/(n - 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-2f02b2035f8f4541","steps":["basis A007318","D"],"edges":["e-c22af348c5e552a2","e-154c3ca700609059"],"text":"A038322 \u2192 basis A007318 \u2192 D"},"headline":"A038322 \u2192 basis A007318 \u2192 D","chain":null}