{"id":"n-cc5ac69d87f94be7","revision":0,"first_index":0,"fingerprint":"f6-6a8026fb20183be6","name":"reversal of basis_map_A008277 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","3","1"],["1","10","25","15","1"],["1","21","140","350","301","63","1"],["1","36","462","2646","6951","7770","3025","255","1"],["1","55","1155","11880","63987","179487","246730","145750","28501","1023","1"],["1","78","2431","39325","359502","1899612","5715424","9321312","7508501","2532530","261625","4095","1"],["1","105","4550","106470","1479478","12662650","67128490","216627840","408741333","420693273","210766920","42355950","2375101","16383","1"],["1","136","7820","249900","4910178","62022324","512060978","2758334150","9528822303","20415995028","25708104786","17505749898","5652751651","694337290","21457825","65535","1"],["1","171","12597","527136","13916778","243577530","2892439160","23466951300","129413217791","477297033785","1144614626805","1709751003480","1492924634839","693081601779","147589284710","11259666950","193448101","262143","1"],["1","210","19285","1023435","34952799","809944464","13087462580","149304004500","1204909218331","6833042030178","26826851689001","71187132291275","123272476465204","132511015347084","82310957214948","26585679462804","3791262568401","181509070050","1742343625","1048575","1"],["1","253","28336","1859550","79781779","2364885369","49916988803","762361127264","8479404429331","68629175807115","401282560341390","1672162773483930","4864251308951100","9593401297313460","12320068811796900","9741955019900400","4382641999117305","998969857983405","96416888184100","2916342574750","15686335501","4194303","1"],["1","300","40250","3200450","168519505","6220194750","166218969675","3275678594925","48063331393110","526655161695960","4299394655347200","25958110360896000","114485073343744260","362262620784874680","802355904438462660","1203163392175387500","1167921451092973005","690223721118368580","227832482998716310","37026417000002430","2436684974110751","46771289738810","141197991025","16777215","1"],["1","351","55575","5265000","333832005","15015551265","495564056130","12246296312250","229268487458010","3270191625210510","35569317763922670","294063066070824960","1834634071262848260","8541149231801585700","29206898819153109600","71823166587281982600","123519417123830092365","143197070509423605675","106563273280541795575","47628831813556336200","11647571772911241531","1359801318005044551","61338207158409090","749329038535350","1270865805301","67108863","1"],["1","406","74907","8336601","626551380","33738295500","1347860993700","40823077538100","949910385013590","17110181160972900","239332331869053150","2598531274376323650","21818248085373723570","140694950355081071520","689692892575539953400","2534474684137526739000","6855064482242755179765","13326679652926121224470","18059551225961878690915","16392038075086211019625","9452962848327254398506","3224318613979279184316","588469772213874823272","49628317055962639176","1540200411172850701","11998160744311570","11438127792025","268435455","1"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","5","52","877","21147","678570","27644437","1382958545","82864869804","5832742205057","474869816156751","44152005855084346","4638590332229999353","545717047936059989389","71339801938860275191172"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1","1"],["0","4","-4"],["0","0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","3"],["0","0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191397,"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-44293a9d609f948d"]},{"id":191398,"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-44293a9d609f948d"]},{"id":191399,"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-44293a9d609f948d"]}],"region_exact":{},"incoming_edges":[{"id":"e-44293a9d609f948d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":2}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-cc5ac69d87f94be7","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"a9e8016174f512737b42c87a89ec7db426ae27b62272db0830ca97111c8c74df","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-64a8f6054296172d","role":"sequence","name":"basis_map_A008277 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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-29f5913b62099c89","kind":"fingerprint_related","node_a_id":"n-64a8f6054296172d","node_b_id":"n-cc5ac69d87f94be7","fingerprint":"f6-6a8026fb20183be6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a8026fb20183be6"},"other_id":"n-64a8f6054296172d","other":{"id":"n-64a8f6054296172d","name":"basis_map_A008277 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-6a8026fb20183be6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3","1"],["1","15","25","10","1"]],"card_rows":[["1"],["1","3","1"],["1","15","25","10","1"],["1","63","301","350","140","21","1"]],"row_sums_preview":["1","5","52","877","21147","678570","27644437","1382958545"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A008277"],"edges":["e-7ccdf7ec12150e18"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008277","chain":null}},{"id":"l-706dd4b973618051","kind":"fingerprint_related","node_a_id":"n-cc5ac69d87f94be7","node_b_id":"n-83e65651eddead61","fingerprint":"f6-6a8026fb20183be6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a8026fb20183be6"},"other_id":"n-83e65651eddead61","other":{"id":"n-83e65651eddead61","name":"basis_map_A008277 of theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-6a8026fb20183be6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","27","9"],["25","375","625","250","25"]],"card_rows":[["1"],["9","27","9"],["25","375","625","250","25"],["49","3087","14749","17150","6860","1029","49"]],"row_sums_preview":["1","45","1300","42973","1712907","82106970","4671909853","311165672625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2","basis A008277"],"edges":["e-5ac7ff393747d6fc","e-43c09f07c4e1e68b"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A008277"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A008277","chain":null}},{"id":"l-f36c2ac192551f7d","kind":"fingerprint_related","node_a_id":"n-cc5ac69d87f94be7","node_b_id":"n-03edc5c28196c3db","fingerprint":"f6-6a8026fb20183be6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a8026fb20183be6"},"other_id":"n-03edc5c28196c3db","other":{"id":"n-03edc5c28196c3db","name":"basis_map_A008277 of theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-6a8026fb20183be6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","9","3"],["5","75","125","50","5"]],"card_rows":[["1"],["3","9","3"],["5","75","125","50","5"],["7","441","2107","2450","980","147","7"]],"row_sums_preview":["1","15","260","6139","190323","7464270","359377681","20744378175"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["\u03b8+1","basis A008277"],"edges":["e-5ff024f1a04a7dc8","e-df664ce7639b6c95"],"text":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 basis A008277"},"headline":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 basis A008277","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: 1;\n  1: 1,   3,     1;\n  2: 1,  10,    25,     15,        1;\n  3: 1,  21,   140,    350,      301,        63,          1;\n  4: 1,  36,   462,   2646,     6951,      7770,       3025,         255,            1;\n  5: 1,  55,  1155,  11880,    63987,    179487,     246730,      145750,        28501,         1023,             1;\n  6: 1,  78,  2431,  39325,   359502,   1899612,    5715424,     9321312,      7508501,      2532530,        261625,          4095,             1;\n  7: 1, 105,  4550, 106470,  1479478,  12662650,   67128490,   216627840,    408741333,    420693273,     210766920,      42355950,       2375101,        16383,            1;\n  8: 1, 136,  7820, 249900,  4910178,  62022324,  512060978,  2758334150,   9528822303,  20415995028,   25708104786,   17505749898,    5652751651,    694337290,     21457825,       65535,         1;\n  9: 1, 171, 12597, 527136, 13916778, 243577530, 2892439160, 23466951300, 129413217791, 477297033785, 1144614626805, 1709751003480, 1492924634839, 693081601779, 147589284710, 11259666950, 193448101, 262143, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-1)*x+(4*n^2-4*n+1)*x^2)*P(n-1,x) - (2*x^2+(4*n-3)*x^3)*d/dx P(n-1,x) + x^4*d^2/dx^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[(1 + (4*n - 1)*x + (4*n^2 - 4*n + 1)*x^2)*P[n - 1, x] - (2*x^2 + (4*n - 3)*x^3)*D[P[n - 1, x], x] + x^4*D[P[n - 1, x], {x, 2}]];\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] = (1 + (4*n - 1)*x + (4*n^2 - 4*n + 1)*x^2)*v[i - 1] - (2*x^2 + (4*n - 3)*x^3)*deriv(v[i - 1], x) + x^4*deriv(deriv(v[i - 1], x), 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-cdb0db4ad473ee28","steps":["basis A008277","reverse"],"edges":["e-7ccdf7ec12150e18","e-44293a9d609f948d"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008277 \u2192 reverse"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008277 \u2192 reverse","chain":null}