{"id":"n-24ae9d775ee03472","revision":0,"first_index":0,"fingerprint":"f6-9024966cc99eeda3","name":"mobius_x_over_x_plus_one of Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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"],["1","25","25"],["1","209","621","414"],["1","1681","11378","19394","9697"],["1","13457","187269","614526","734520","293808"],["1","107665","2941922","16607289","35650582","32816325","10938775"],["1","861329","45192021","413836192","1421447090","2260348482","1691769646","483362756"],["1","6890641","686385938","9848815669","50877042503","124381763149","156858256961","98912287924","24728071981"],["1","55125137","10363834149","227978493176","1705170190465","6016901731428","11372802854840","11875042529380","6469252713855","1437611714190"],["1","441001105","156001872962","5186201132309","54767187277782","268188649052562","712882698460520","1096124132222730","976164967502310","468089289631085","93617857926217"],["1","3528008849","2344382980341","116665387593192","1709199453390060","11311339025011179","40630990552200405","86294773275503256","111706468808680440","86703692718226065","37128019789778247","6750549052686954"],["1","28224070801","35200583792498","2605243754854149","52287483483992053","458903862220974881","2167912828524343446","6109369747504572194","10780965206960233886","12042517548568008960","8281939211558883527","3203692201006517310","533948700167752885"],["1","225792566417","528287469586629","57895007324395576","1577090695791429695","18101944598361740070","110290838298071647448","401220248137340202412","926981354266260439023","1397403113806635424700","1372170195808289658352","847054436898303994308","298784473375671119122","45966842057795556788"],["1","1806340531345","7926541745392802","1282389362305131829","47085698456783085932","699366975505437456784","5415856336376399991392","24932197008817379836601","73358110826213397756803","143316950175749026286694","188415050679108566048414","165008771373585455204777","92394380171786018481561","29950577665933127360353","4278653952276161051479"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","51","1245","42151","1843581","99062559","6316817517","466299514767","39115179186621","3675177426079583","382354045974058989","43634177262557996591","5419608961463107760541","727818291181518143616767"],"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"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","7","21","21","7"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","7","21","21","7"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191297,"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-6b73c0246f6b30b7"]},{"id":191298,"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-6b73c0246f6b30b7"]},{"id":191299,"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-6b73c0246f6b30b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-6b73c0246f6b30b7","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-24ae9d775ee03472","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":"41d6cc491e09ebb5a93cf782c19cc1a0f93c454a87bffdac8bde17d437480175","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-2d05bfc763190112","role":"sequence","name":"Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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-526ca225333fa304","kind":"fingerprint_related","node_a_id":"n-07cdb61aa05a9163","node_b_id":"n-24ae9d775ee03472","fingerprint":"f6-9024966cc99eeda3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9024966cc99eeda3"},"other_id":"n-07cdb61aa05a9163","other":{"id":"n-07cdb61aa05a9163","name":"shift_x_plus_one of Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","first_index":0,"fingerprint":"f6-9024966cc99eeda3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["25","25","1"]],"card_rows":[["1"],["2","1"],["25","25","1"],["414","621","209","1"]],"row_sums_preview":["1","3","51","1245","42151","1843581","99062559","6316817517"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2d05bfc763190112","steps":["x \u21a6 x+1"],"edges":["e-65f799b74d70a3de"],"text":"A144445 \u2192 x \u21a6 x+1"},"headline":"A144445 \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: 1,       25,          25;\n  3: 1,      209,         621,          414;\n  4: 1,     1681,       11378,        19394,          9697;\n  5: 1,    13457,      187269,       614526,        734520,        293808;\n  6: 1,   107665,     2941922,     16607289,      35650582,      32816325,       10938775;\n  7: 1,   861329,    45192021,    413836192,    1421447090,    2260348482,     1691769646,      483362756;\n  8: 1,  6890641,   686385938,   9848815669,   50877042503,  124381763149,   156858256961,    98912287924,   24728071981;\n  9: 1, 55125137, 10363834149, 227978493176, 1705170190465, 6016901731428, 11372802854840, 11875042529380, 6469252713855, 1437611714190;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (7*x+21*x^2+21*x^3+7*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (7*x+21*x^2+21*x^3+7*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*x + 21*x^2 + 21*x^3 + 7*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (7*x + 21*x^2 + 21*x^3 + 7*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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*x + 21*x^2 + 21*x^3 + 7*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (7*x + 21*x^2 + 21*x^3 + 7*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-2d05bfc763190112","steps":["x \u21a6 x/(x+1)"],"edges":["e-6b73c0246f6b30b7"],"text":"A144445 \u2192 x \u21a6 x/(x+1)"},"headline":"A144445 \u2192 x \u21a6 x/(x+1)","chain":null}