{"id":"n-e763351afcabc7a7","revision":0,"first_index":0,"fingerprint":"f6-570ab0ccd1213152","name":"mobius_x_over_x_plus_one of basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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":false,"rows":[["1"],["1","2"],["2","8","7"],["6","36","63","34"],["24","192","504","544","209"],["120","1200","4200","6800","5225","1546"],["720","8640","37800","81600","94050","55656","13327"],["5040","70560","370440","999600","1536150","1363572","653023","130922"],["40320","645120","3951360","12794880","24578400","29089536","20896736","8379008","1441729"],["362880","6531840","45722880","172730880","398170080","589063104","564211872","339349824","116780049","17572114"],["3628800","72576000","571536000","2467584000","6636168000","11781262080","14105296800","11311660800","5839002450","1757211400","234662231"],["39916800","878169600","7683984000","37322208000","114710904000","237588785280","341348182560","342177739200","235506432150","106311289700","28394129951","3405357682"],["479001600","11496038400","110649369600","597155328000","2064796272000","4887540725760","8192356381440","9854718888960","8478231557400","5102941905600","2044377356472","490371506208","53334454417"],["6227020800","161902540800","1699976678400","10091925043200","38772285552000","103249297831680","197786889780480","277574582039040","286564226640120","215599295511600","115166591081256","41436392274576","9013522796473","896324308634"],["87178291200","2440992153600","27766285747200","179819755315200","759936796819200","2248540263889920","4845778799621760","7772088297093120","9361098070243920","8451492384054720","5643162962981544","2707177628605632","883325234054354","175679564492264","16083557845279"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","139","1473","19091","291793","5129307","101817089","2250495523","54780588561","1455367098923","41888448785857","1298019439099059","43074477771208913"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","2","3","1"],["1","1","-1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180837,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-b400a5696574ba31"]},{"id":180838,"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-b400a5696574ba31"]},{"id":180839,"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-b400a5696574ba31"]}],"region_exact":{},"incoming_edges":[{"id":"e-b400a5696574ba31","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-e763351afcabc7a7","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"c68a4a9142f274b91543aac4359754c5b216452a81baa03da16cf77bb79230aa","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8ab17db79dad1880","role":"sequence","name":"basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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-16a1cfeca3fc920b","kind":"fingerprint_related","node_a_id":"n-12211825af356650","node_b_id":"n-e763351afcabc7a7","fingerprint":"f6-570ab0ccd1213152","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-570ab0ccd1213152"},"other_id":"n-12211825af356650","other":{"id":"n-12211825af356650","name":"shift_x_plus_one of T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","first_index":0,"fingerprint":"f6-570ab0ccd1213152","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["7","8","2"]],"card_rows":[["1"],["2","1"],["7","8","2"],["34","63","36","6"]],"row_sums_preview":["1","3","17","139","1473","19091","291793","5129307"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-47fe395e9f7d5f86","steps":["x \u21a6 x+1"],"edges":["e-54b278190ec3c985"],"text":"A144084 \u2192 x \u21a6 x+1"},"headline":"A144084 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-424d3bb9afb073ec","kind":"fingerprint_related","node_a_id":"n-e763351afcabc7a7","node_b_id":"n-6e47c52ce477d423","fingerprint":"f6-570ab0ccd1213152","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-570ab0ccd1213152"},"other_id":"n-6e47c52ce477d423","other":{"id":"n-6e47c52ce477d423","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","first_index":1,"fingerprint":"f6-570ab0ccd1213152","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["28","32","8"],["272","504","288","48"],["3344","8704","8064","3072","384"]],"card_rows":[["28","32","8"],["272","504","288","48"],["3344","8704","8064","3072","384"],["49472","167200","217600","134400","38400","3840"]],"row_sums_preview":["68","1112","23568","610912","18674752","656551296","26065174784","1152253707776"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["x \u21a6 x+1"],"edges":["e-ed5af84bfccfc67f"],"text":"A395972 \u2192 x \u21a6 x+1"},"headline":"A395972 \u2192 x \u21a6 x+1","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,       2;\n  2:      2,       8,        7;\n  3:      6,      36,       63,        34;\n  4:     24,     192,      504,       544,       209;\n  5:    120,    1200,     4200,      6800,      5225,      1546;\n  6:    720,    8640,    37800,     81600,     94050,     55656,     13327;\n  7:   5040,   70560,   370440,    999600,   1536150,   1363572,    653023,    130922;\n  8:  40320,  645120,  3951360,  12794880,  24578400,  29089536,  20896736,   8379008,   1441729;\n  9: 362880, 6531840, 45722880, 172730880, 398170080, 589063104, 564211872, 339349824, 116780049, 17572114;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+(n+2)*x-(n-3)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx 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[((n + (n + 2)*x - (n - 3)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 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] = ((n + (n + 2)*x - (n - 3)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 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-b9fbfb57d9269e2f","steps":["basis A008277","x \u21a6 x/(x+1)"],"edges":["e-07cfebc08bea7e3c","e-b400a5696574ba31"],"text":"A132393 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A132393 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)","chain":null}
