{"id":"n-c4d0a0348e31ddee","revision":0,"first_index":0,"fingerprint":"f6-3a0db69ffdaa907b","name":"mobius_x_over_x_plus_one of basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["9","24","16"],["18","75","104","48"],["36","204","433","408","144"],["72","516","1478","2115","1512","432"],["144","1248","4504","8664","9369","5400","1296"],["288","2928","12752","30840","44730","38907","18792","3888"],["576","6720","34288","99936","181980","212004","154305","64152","11664"],["1152","15168","88736","302736","663768","969948","944622","591219","215784","34992"],["2304","33792","222976","871680","2235744","3931200","4799088","4016304","2205225","717336","104976"],["4608","74496","547328","2412288","7086528","14569632","21391776","22429872","16459362","8050347","2361960","314928"],["9216","162816","1318144","6466560","21409920","50398848","86492448","109035072","100208340","65478780","28874961","7715736","944784"],["18432","353280","3124736","16887552","62219520","165027456","324181440","477547488","527521896","431582580","254186262","102056355","25036776","2834352"],["36864","761856","7309312","43149312","175101696","516713472","1143445248","1927639296","2487686256","2445730848","1803120264","966671496","356242617","80779032","8503056"],["73728","1634304","16904192","108226560","479651328","1558732032","3837030912","7285614336","10758290400","12354520464","10943433072","7342703784","3612499722","1230285915","259343208","25509168"],["147456","3489792","38711296","267165696","1283982336","4556418048","12350257920","26082321408","43373423808","56983912128","58950427536","47515706784","29253110796","13298070996","4209544161","829047960","76527504"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["49","245","1225","6125","30625","153125","765625","3828125","19140625","95703125","478515625","2392578125","11962890625","59814453125","299072265625"],"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":[["9","24","16"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["2","5","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175909,"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-da236774fda5a2ed"]},{"id":175910,"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-da236774fda5a2ed"]},{"id":175911,"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-da236774fda5a2ed"]}],"region_exact":{},"incoming_edges":[{"id":"e-da236774fda5a2ed","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-c4d0a0348e31ddee","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":"b6345721cc9b59497ea1ac5b0d92ec770058755f7640ec90402a2044ad8bc0fb","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-39578f69b35e8349","role":"sequence","name":"basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    9,    24,     16;\n  1:   18,    75,    104,      48;\n  2:   36,   204,    433,     408,     144;\n  3:   72,   516,   1478,    2115,    1512,      432;\n  4:  144,  1248,   4504,    8664,    9369,     5400,     1296;\n  5:  288,  2928,  12752,   30840,   44730,    38907,    18792,     3888;\n  6:  576,  6720,  34288,   99936,  181980,   212004,   154305,    64152,    11664;\n  7: 1152, 15168,  88736,  302736,  663768,   969948,   944622,   591219,   215784,   34992;\n  8: 2304, 33792, 222976,  871680, 2235744,  3931200,  4799088,  4016304,  2205225,  717336,  104976;\n  9: 4608, 74496, 547328, 2412288, 7086528, 14569632, 21391776, 22429872, 16459362, 8050347, 2361960, 314928;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2+5*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 16*x^2.","mathematica":"P[0, x_] := 9 + 24*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + 5*x + 3*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] = 9 + 24*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 5*x + 3*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-f48bed27dbc29c5f","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-3fc4d53dea6f488e","e-da236774fda5a2ed"],"text":"A391533 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A391533 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}
