{"id":"n-5364558beebadbdf","revision":0,"first_index":0,"fingerprint":"f6-620515e8c92e1339","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Pascal-(1,3,1) array.","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"],["2","5"],["7","35","37"],["20","150","348","245"],["61","610","2166","3205","1669"],["182","2275","10916","24995","27214","11285"],["547","8205","49587","153995","258477","222135","76453"],["1640","28700","209256","821950","1874488","2478660","1759424","517685"],["4921","98420","840412","3994430","11539126","20721260","22575256","13644905","3505861"],["14762","332145","3250872","18140010","63514188","144554850","213708840","197854425","104114610","23741525"],["44287","1107175","12219057","78304890","322347198","889864470","1667206110","2092471725","1683595575","784382875","160777957"],["132860","3653650","44885924","324873915","1537872024","4995522420","11355354096","18054828075","19674216060","13993940450","5848968484","1088788085"],["398581","11957430","161836338","1305664075","6988357251","26124865020","69906072048","134850307095","186066908415","179075704150","114118863114","43246682445","7373275909"],["1195742","38861615","574516668","5112264245","30530390762","129144650895","397889832072","903672039135","1512494670762","1842238949525","1587162851900","916015857375","317499407126","49931836565"],["3587227","125552945","2012994847","19585138285","129114171683","609809178325","2126954397075","5563433620875","10963177733445","16193385676675","17646815503997","13758277418095","7255324522897","2316920036735","338138493733"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","79","763","7711","76867","769399","7691803","76924591","769226227","7692321319","76923036043","769230891871","7692307324387","76923078026839"],"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":[["0","0","0","0"],["9","27","27","9"]]},"terms":[{"coeff":{"coeffs":[["233","1749","5250","8170","7005","3153","584"],["-315","-2370","-7233","-11505","-10092","-4641","-876"],["100","720","2172","3488","3132","1488","292"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["184","1939","8103","17870","22790","16959","6859","1168"],["-80","-848","-3597","-8092","-10558","-8052","-3341","-584"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["16","224","1273","3857","6910","7586","5029","1853","292"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182261,"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-82367c66d3bca2eb"]},{"id":182262,"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-82367c66d3bca2eb"]},{"id":182263,"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-82367c66d3bca2eb"]}],"region_exact":{},"incoming_edges":[{"id":"e-82367c66d3bca2eb","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-5364558beebadbdf","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":"1fa297f23a929e4537a5daef5e95d1ba9935605be64c8e383b648461bac6ea94","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-9813839be9bee901","role":"sequence","name":"basis_map_A013610 of Pascal-(1,3,1) array.","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:     2,      5;\n  2:     7,     35,      37;\n  3:    20,    150,     348,      245;\n  4:    61,    610,    2166,     3205,     1669;\n  5:   182,   2275,   10916,    24995,    27214,     11285;\n  6:   547,   8205,   49587,   153995,   258477,    222135,     76453;\n  7:  1640,  28700,  209256,   821950,  1874488,   2478660,   1759424,    517685;\n  8:  4921,  98420,  840412,  3994430, 11539126,  20721260,  22575256,  13644905,   3505861;\n  9: 14762, 332145, 3250872, 18140010, 63514188, 144554850, 213708840, 197854425, 104114610, 23741525;\n","recurrence_sentence":"The row polynomials satisfy (9*n+27*n*x+27*n*x^2+9*n*x^3)*P(n,x) = (100*n^2-315*n+233+(720*n^2-2370*n+1749)*x+(2172*n^2-7233*n+5250)*x^2+(3488*n^2-11505*n+8170)*x^3+(3132*n^2-10092*n+7005)*x^4+(1488*n^2-4641*n+3153)*x^5+(292*n^2-876*n+584)*x^6)*P(n-1,x) - (80*n-184+(848*n-1939)*x+(3597*n-8103)*x^2+(8092*n-17870)*x^3+(10558*n-22790)*x^4+(8052*n-16959)*x^5+(3341*n-6859)*x^6+(584*n-1168)*x^7)*d/dx P(n-1,x) + (16+224*x+1273*x^2+3857*x^3+6910*x^4+7586*x^5+5029*x^6+1853*x^7+292*x^8)*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[Together[((100*n^2 - 315*n + 233 + (720*n^2 - 2370*n + 1749)*x + (2172*n^2 - 7233*n + 5250)*x^2 + (3488*n^2 - 11505*n + 8170)*x^3 + (3132*n^2 - 10092*n + 7005)*x^4 + (1488*n^2 - 4641*n + 3153)*x^5 + (292*n^2 - 876*n + 584)*x^6)*P[n - 1, x] - (80*n - 184 + (848*n - 1939)*x + (3597*n - 8103)*x^2 + (8092*n - 17870)*x^3 + (10558*n - 22790)*x^4 + (8052*n - 16959)*x^5 + (3341*n - 6859)*x^6 + (584*n - 1168)*x^7)*D[P[n - 1, x], x] + (16 + 224*x + 1273*x^2 + 3857*x^3 + 6910*x^4 + 7586*x^5 + 5029*x^6 + 1853*x^7 + 292*x^8)*D[P[n - 1, x], {x, 2}])/(9*n + 27*n*x + 27*n*x^2 + 9*n*x^3)]];\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] = ((100*n^2 - 315*n + 233 + (720*n^2 - 2370*n + 1749)*x + (2172*n^2 - 7233*n + 5250)*x^2 + (3488*n^2 - 11505*n + 8170)*x^3 + (3132*n^2 - 10092*n + 7005)*x^4 + (1488*n^2 - 4641*n + 3153)*x^5 + (292*n^2 - 876*n + 584)*x^6)*v[i - 1] - (80*n - 184 + (848*n - 1939)*x + (3597*n - 8103)*x^2 + (8092*n - 17870)*x^3 + (10558*n - 22790)*x^4 + (8052*n - 16959)*x^5 + (3341*n - 6859)*x^6 + (584*n - 1168)*x^7)*deriv(v[i - 1], x) + (16 + 224*x + 1273*x^2 + 3857*x^3 + 6910*x^4 + 7586*x^5 + 5029*x^6 + 1853*x^7 + 292*x^8)*deriv(deriv(v[i - 1], x), x))/(9*n + 27*n*x + 27*n*x^2 + 9*n*x^3)); 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-d8f9ad9ce0e039d0","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-a15ab36c8834192d","e-82367c66d3bca2eb"],"text":"A081578 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A081578 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}