{"id":"n-aeb498ce26faf029","revision":0,"first_index":0,"fingerprint":"f6-3a176e70253767ae","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^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"],["20","38"],["400","1520","1444"],["8000","45600","86640","54872"],["160000","1216000","3465600","4389760","2085136"],["3200000","30400000","115520000","219488000","208513600","79235168"],["64000000","729600000","3465600000","8779520000","12510816000","9508220160","3010936384"],["1280000000","17024000000","97036800000","307283200000","583838080000","665575411200","421531093760","114415582592"],["25600000000","389120000000","2587648000000","9833062400000","23353523200000","35497355264000","33722487500800","18306493214720","4347792138496"],["512000000000","8755200000000","66539520000000","294991872000000","840726835200000","1597380986880000","2023349250048000","1647584389324800","782602584929280","165216101262848"],["10240000000000","194560000000000","1663488000000000","8428339200000000","28024227840000000","63895239475200000","101167462502400000","109838959288320000","78260258492928000","33043220252569600","6278211847988224"],["204800000000000","4280320000000000","40663040000000000","231779328000000000","880761446400000000","2342825447424000000","4451368350105600000","6041142760857600000","5739085622814720000","3634754227782656000","1381206606557409280","238572050223552512"],["4096000000000000","93388800000000000","975912960000000000","6180782080000000000","26422843392000000000","80325443911680000000","178054734004224000000","289974852521164800000","344345137368883200000","290780338222612480000","165744792786889113600","57257292053652602880","9065737908494995456"],["81920000000000000","2023424000000000000","23067033600000000000","160700334080000000000","763326586880000000000","2610576927129600000000","6613461548728320000000","12565576942583808000000","17905947143181926400000","18900721984469811200000","14364548708197056512000","7443447966974838374400","2357091856208698818560","344498040522809827328"],["1638400000000000000","43581440000000000000","538230784000000000000","4090553958400000000000","21373144432640000000000","81217948844032000000000","231471154205491200000000","502623077703352320000000","835610866681823232000000","1058440431130309427200000","1005518409573793955840000","694721810250984914944000","329992859869217834598400","96459451346386751651840","13090925539866773438464"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","58","3364","195112","11316496","656356768","38068692544","2207984167552","128063081718016","7427658739644928","430804206899405824","24986644000165537792","1449225352009601191936","84055070416556869132288","4875194084160298409672704"],"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":[["20","58","38"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193060,"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-31713855ed754395"]},{"id":193061,"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-31713855ed754395"]},{"id":193062,"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-31713855ed754395"]}],"region_exact":{},"incoming_edges":[{"id":"e-31713855ed754395","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-aeb498ce26faf029","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":"f63726ab27d1b0d12e861374f27b64bb0b6d5d63571eb26e40b12b1d03e25728","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-76276d62adb9bc90","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^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:           20,            38;\n  2:          400,          1520,           1444;\n  3:         8000,         45600,          86640,           54872;\n  4:       160000,       1216000,        3465600,         4389760,         2085136;\n  5:      3200000,      30400000,      115520000,       219488000,       208513600,         79235168;\n  6:     64000000,     729600000,     3465600000,      8779520000,     12510816000,       9508220160,       3010936384;\n  7:   1280000000,   17024000000,    97036800000,    307283200000,    583838080000,     665575411200,     421531093760,     114415582592;\n  8:  25600000000,  389120000000,  2587648000000,   9833062400000,  23353523200000,   35497355264000,   33722487500800,   18306493214720,   4347792138496;\n  9: 512000000000, 8755200000000, 66539520000000, 294991872000000, 840726835200000, 1597380986880000, 2023349250048000, 1647584389324800, 782602584929280, 165216101262848;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (20+58*x+38*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[((20 + 58*x + 38*x^2)*P[n - 1, 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] = ((20 + 58*x + 38*x^2)*v[i - 1])/(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-4e096b5929fd3f2e","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-66940ce826511afb","e-31713855ed754395"],"text":"A038323 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038323 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}