{"id":"n-80bb1878a7313c9f","revision":0,"first_index":1,"fingerprint":"f6-357ed7a70cf3dab3","name":"reversal of basis_map_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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":false,"rows":[["1"],["2","3"],["4","20","15"],["8","84","210","105"],["16","288","1512","2520","945"],["32","880","7920","27720","34650","10395"],["64","2496","34320","205920","540540","540540","135135"],["128","6720","131040","1201200","5405400","11351340","9459450","2027025"],["256","17408","456960","5940480","40840800","147026880","257297040","183783600","34459425"],["512","43776","1488384","26046720","253955520","1396755360","4190266080","6285399120","3928374450","654729075"],["1024","107520","4596480","104186880","1367452800","10666131840","48886437600","125707982400","164991726900","91662070500","13749310575"],["2048","259072","13601280","387636480","6589820160","69193111680","449755225920","1766895530400","3975514943400","4638100767300","2319050383650","316234143225"],["4096","614400","38860800","1360128000","29072736000","395389209600","3459655584000","19275223968000","66258582390000","132517164780000","139143023019000","63246828645000","7905853580625"],["8192","1437696","107827200","4546713600","119351232000","2040906067200","23130268761600","173477015712000","845700451596000","2584084713210000","4651352483778000","4439927370879000","1849969737866250","213458046676875"],["16384","3325952","291852288","14592614400","461491430400","9691320038400","138101310547200","1341555588172800","8803958547384000","38150487038664000","104913839356326000","171677191673988000","150217542714739500","57775977967207500","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","39","407","5281","81597","1459015","29582303","669822849","16737058997","457140004519","13541735424615","432232193519521","14783265001765613","539219105249936199"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-5"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187601,"position":0,"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-41f6ca9a401302a7"]},{"id":187602,"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-41f6ca9a401302a7"]},{"id":187603,"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-41f6ca9a401302a7"]}],"region_exact":{},"incoming_edges":[{"id":"e-41f6ca9a401302a7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-80bb1878a7313c9f","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":"e0fdd9b52da5e98b5ee223a722d91cbf0220f421c19b5db0913522770876a5f1","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-deb68900633836aa","role":"sequence","name":"basis_map_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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":[{"id":"l-0b1799bef0daeda2","kind":"fingerprint_related","node_a_id":"n-80bb1878a7313c9f","node_b_id":"n-8090c1757ab2328b","fingerprint":"f6-357ed7a70cf3dab3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-357ed7a70cf3dab3"},"other_id":"n-8090c1757ab2328b","other":{"id":"n-8090c1757ab2328b","name":"basis_map_A008277 of basis_map_A013609 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-357ed7a70cf3dab3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["45","60","12"]],"card_rows":[["1"],["3","2"],["45","60","12"],["1575","3150","1260","120"]],"row_sums_preview":["1","5","117","6105","554505","77109165","15166460925","3997604515905"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013609","basis A008277"],"edges":["e-2613ba1fbc2604f1","e-bc3c6ed364a44e85"],"text":"A204420 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A204420 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-4cbdf1f9e4518840","kind":"fingerprint_related","node_a_id":"n-80bb1878a7313c9f","node_b_id":"n-42e8691f6ca17cbd","fingerprint":"f6-357ed7a70cf3dab3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-357ed7a70cf3dab3"},"other_id":"n-42e8691f6ca17cbd","other":{"id":"n-42e8691f6ca17cbd","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","first_index":0,"fingerprint":"f6-357ed7a70cf3dab3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["60","80","16"]],"card_rows":[["1"],["6","4"],["60","80","16"],["840","1680","672","64"]],"row_sums_preview":["1","10","156","3256","84496","2611104","93376960","3786534784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013611","basis A008277"],"edges":["e-403803ced55fc794","e-cf73544ec8921c50"],"text":"A290319 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A290319 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-d9ba4b39414c8219","kind":"fingerprint_related","node_a_id":"n-deb68900633836aa","node_b_id":"n-80bb1878a7313c9f","fingerprint":"f6-357ed7a70cf3dab3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-357ed7a70cf3dab3"},"other_id":"n-deb68900633836aa","other":{"id":"n-deb68900633836aa","name":"basis_map_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","first_index":1,"fingerprint":"f6-357ed7a70cf3dab3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["15","20","4"]],"card_rows":[["1"],["3","2"],["15","20","4"],["105","210","84","8"]],"row_sums_preview":["1","5","39","407","5281","81597","1459015","29582303"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A008277"],"edges":["e-0664c5d711618d77"],"text":"A161198 \u2192 basis A008277"},"headline":"A161198 \u2192 basis A008277","chain":null}}],"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:   1;\n   2:   2,     3;\n   3:   4,    20,      15;\n   4:   8,    84,     210,      105;\n   5:  16,   288,    1512,     2520,       945;\n   6:  32,   880,    7920,    27720,     34650,      10395;\n   7:  64,  2496,   34320,   205920,    540540,     540540,     135135;\n   8: 128,  6720,  131040,  1201200,   5405400,   11351340,    9459450,    2027025;\n   9: 256, 17408,  456960,  5940480,  40840800,  147026880,  257297040,  183783600,   34459425;\n  10: 512, 43776, 1488384, 26046720, 253955520, 1396755360, 4190266080, 6285399120, 3928374450, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(4*n-5)*x)*P(n-1,x) - 2*x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + (4*n - 5)*x)*P[n - 1, x] - 2*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2 + (4*n - 5)*x)*v[i - 1] - 2*x^2*deriv(v[i - 1], x)); 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-7c9c430277778c10","steps":["basis A008277","reverse"],"edges":["e-0664c5d711618d77","e-41f6ca9a401302a7"],"text":"A161198 \u2192 basis A008277 \u2192 reverse"},"headline":"A161198 \u2192 basis A008277 \u2192 reverse","chain":null}