{"id":"n-f296d09d2ec50000","revision":0,"first_index":0,"fingerprint":"f6-ec2ecf19f26d1b35","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.","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"],["5","10"],["25","110","110"],["125","910","1980","1320"],["625","6710","23870","34320","17160"],["3125","46510","240900","561000","600600","240240"],["15625","310310","2197910","7378800","12698400","10810800","3603600"],["78125","2018110","18798780","85405320","210210000","285885600","201801600","57657600"],["390625","12889910","153790670","908827920","3002159160","5801796000","6508101600","3920716800","980179200"],["1953125","81245710","1218323700","9116111400","38834195400","100048188240","158616057600","151178227200","79394515200","17643225600"],["9765625","507005510","9421968710","87548775600","468017206800","1544160618000","3245499457200","4351995648000","3602158560000","1676106432000","335221286400"],["48828125","3139689310","71530841580","813453829320","5350288944000","21993847075200","58862903299200","104151939091200","120811987296000","88304344128000","36874341504000","6704425728000"],["244140625","19326417110","535252473470","7366000733520","58727500277160","294842515968000","977399642419200","2191629721881600","3341138799398400","3410876589120000","2230897660992000","844757641728000","140792940288000"],["1220703125","118399908910","3959357902500","65351035549800","624305512030200","3770610163560240","15174033806131200","41937950941286400","80692509664166400","107892770636851200","98270120108160000","58134075487488000","20133390461184000","3097444686336000"],["6103515625","722606484710","29017904315510","570320579228400","6468313070419200","46446378804025200","223473524320846800","746039952193536000","1761947791636032000","2962963962870912000","3524014443722572800","2895547609963008000","1563083223077376000","498688594500096000","71241227785728000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","245","4335","82685","1692375","37015445","861855135","21288851885","556132043175","15320646723845","443942250253935","13498983835837085","429796641481257975","14300515089138096245"],"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","2","1"]]},"terms":[{"coeff":{"coeffs":[["5","20","25","10"]]},"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":188059,"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-6c77734a28bd1ce3"]},{"id":188060,"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-6c77734a28bd1ce3"]},{"id":188061,"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-6c77734a28bd1ce3"]}],"region_exact":{},"incoming_edges":[{"id":"e-6c77734a28bd1ce3","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-f296d09d2ec50000","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":"d50c03910eb3c557129e6f56243219b90bb9ce76cbc7c6efddf1c468d2d257bc","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-8919a5e7419ba068","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.","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-f811e7c4c4a5724a","kind":"fingerprint_related","node_a_id":"n-fa0ad7db8c6303f8","node_b_id":"n-f296d09d2ec50000","fingerprint":"f6-ec2ecf19f26d1b35","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ec2ecf19f26d1b35"},"other_id":"n-fa0ad7db8c6303f8","other":{"id":"n-fa0ad7db8c6303f8","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.","first_index":0,"fingerprint":"f6-ec2ecf19f26d1b35","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","5"],["110","110","25"]],"card_rows":[["1"],["10","5"],["110","110","25"],["1320","1980","910","125"]],"row_sums_preview":["1","15","245","4335","82685","1692375","37015445","861855135"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8919a5e7419ba068","steps":["x \u21a6 x+1"],"edges":["e-4edbe7542d501921"],"text":"A257607 \u2192 x \u21a6 x+1"},"headline":"A257607 \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:       5,       10;\n  2:      25,      110,        110;\n  3:     125,      910,       1980,       1320;\n  4:     625,     6710,      23870,      34320,       17160;\n  5:    3125,    46510,     240900,     561000,      600600,       240240;\n  6:   15625,   310310,    2197910,    7378800,    12698400,     10810800,      3603600;\n  7:   78125,  2018110,   18798780,   85405320,   210210000,    285885600,    201801600,     57657600;\n  8:  390625, 12889910,  153790670,  908827920,  3002159160,   5801796000,   6508101600,   3920716800,   980179200;\n  9: 1953125, 81245710, 1218323700, 9116111400, 38834195400, 100048188240, 158616057600, 151178227200, 79394515200, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+20*x+25*x^2+10*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[((5 + 20*x + 25*x^2 + 10*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((5 + 20*x + 25*x^2 + 10*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-8919a5e7419ba068","steps":["x \u21a6 x/(x+1)"],"edges":["e-6c77734a28bd1ce3"],"text":"A257607 \u2192 x \u21a6 x/(x+1)"},"headline":"A257607 \u2192 x \u21a6 x/(x+1)","chain":null}
