{"id":"n-07cdb61aa05a9163","revision":0,"first_index":0,"fingerprint":"f6-9024966cc99eeda3","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.","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","1"],["25","25","1"],["414","621","209","1"],["9697","19394","11378","1681","1"],["293808","734520","614526","187269","13457","1"],["10938775","32816325","35650582","16607289","2941922","107665","1"],["483362756","1691769646","2260348482","1421447090","413836192","45192021","861329","1"],["24728071981","98912287924","156858256961","124381763149","50877042503","9848815669","686385938","6890641","1"],["1437611714190","6469252713855","11875042529380","11372802854840","6016901731428","1705170190465","227978493176","10363834149","55125137","1"],["93617857926217","468089289631085","976164967502310","1096124132222730","712882698460520","268188649052562","54767187277782","5186201132309","156001872962","441001105","1"],["6750549052686954","37128019789778247","86703692718226065","111706468808680440","86294773275503256","40630990552200405","11311339025011179","1709199453390060","116665387593192","2344382980341","3528008849","1"],["533948700167752885","3203692201006517310","8281939211558883527","12042517548568008960","10780965206960233886","6109369747504572194","2167912828524343446","458903862220974881","52287483483992053","2605243754854149","35200583792498","28224070801","1"],["45966842057795556788","298784473375671119122","847054436898303994308","1372170195808289658352","1397403113806635424700","926981354266260439023","401220248137340202412","110290838298071647448","18101944598361740070","1577090695791429695","57895007324395576","528287469586629","225792566417","1"],["4278653952276161051479","29950577665933127360353","92394380171786018481561","165008771373585455204777","188415050679108566048414","143316950175749026286694","73358110826213397756803","24932197008817379836601","5415856336376399991392","699366975505437456784","47085698456783085932","1282389362305131829","7926541745392802","1806340531345","1"]],"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:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","7"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-5","-6"],["7","7"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-7","-7"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191293,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-65f799b74d70a3de"]},{"id":191294,"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-65f799b74d70a3de"]},{"id":191295,"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-65f799b74d70a3de"]},{"id":191296,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-65f799b74d70a3de"]}],"region_exact":{},"incoming_edges":[{"id":"e-65f799b74d70a3de","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-07cdb61aa05a9163","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"b9b91d0d8ce5bb76c160aec0f72b06bae589dad2b72a4494297766a291c1f888","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"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-24ae9d775ee03472","other":{"id":"n-24ae9d775ee03472","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.","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"],["1","2"],["1","25","25"]],"card_rows":[["1"],["1","2"],["1","25","25"],["1","209","621","414"]],"row_sums_preview":["1","3","51","1245","42151","1843581","99062559","6316817517"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"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}}],"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,             1;\n  2:            25,            25,              1;\n  3:           414,           621,            209,              1;\n  4:          9697,         19394,          11378,           1681,             1;\n  5:        293808,        734520,         614526,         187269,         13457,             1;\n  6:      10938775,      32816325,       35650582,       16607289,       2941922,        107665,            1;\n  7:     483362756,    1691769646,     2260348482,     1421447090,     413836192,      45192021,       861329,           1;\n  8:   24728071981,   98912287924,   156858256961,   124381763149,   50877042503,    9848815669,    686385938,     6890641,        1;\n  9: 1437611714190, 6469252713855, 11875042529380, 11372802854840, 6016901731428, 1705170190465, 227978493176, 10363834149, 55125137, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+7*x)*P(n-2,x) + (7*n-5+(7*n-6)*x)*P(n-1,x) - (7*x+7*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(7 + 7*x)*P[n - 2, x] + (7*n - 5 + (7*n - 6)*x)*P[n - 1, x] - (7*x + 7*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (7 + 7*x)*v[i - 2] + (7*n - 5 + (7*n - 6)*x)*v[i - 1] - (7*x + 7*x^2)*deriv(v[i - 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-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}