{"id":"n-1b09cdcaf124bad8","revision":0,"first_index":0,"fingerprint":"f6-b7480fed0cfd4971","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","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":[["625","350","49"],["15625","13125","3675","343"],["390625","437500","183750","34300","2401"],["9765625","13671875","7656250","2143750","300125","16807"],["244140625","410156250","287109375","107187500","22509375","2521050","117649"],["6103515625","11962890625","10048828125","4689453125","1313046875","220591875","20588575","823543"],["152587890625","341796875000","334960937500","187578125000","65652343750","14706125000","2058857500","164708600","5764801"],["3814697265625","9613037109375","10766601562500","7034179687500","2954355468750","827219531250","154414312500","18529717500","1297080225","40353607"],["95367431640625","267028808593750","336456298828125","251220703125000","123098144531250","41360976562500","9650894531250","1544143125000","162135028125","10088401750","282475249"],["2384185791015625","7343292236328125","10280609130859375","8635711669921875","4835998535156250","1895711425781250","530799199218750","106159839843750","14862377578125","1387155240625","77680693475","1977326743"],["59604644775390625","200271606445312500","308418273925781250","287857055664062500","181349945068359375","81244775390625000","26539959960937500","6369590390625000","1114678318359375","138715524062500","11652104021250","593198022900","13841287201"],["1490116119384765625","5424022674560546875","9112358093261718750","9355354309082031250","6548748016357421875","3300569000244140625","1232212426757812500","345019479492187500","72454090693359375","11270636330078125","1262311268968750","96394678721250","4498418340325","96889010407"],["37252902984619140625","146031379699707031250","265777111053466796875","297670364379882812500","229206180572509765625","128355461120605468750","53909293670654296875","17250973974609375000","4226488623779296875","788944543105468750","110452236034765625","11246045850812500","787223209556875","33911153642450","678223072849"],["931322574615478515625","3911554813385009765625","7666647434234619140625","9302198886871337890625","7813847064971923828125","4813329792022705078125","2246220569610595703125","808639405059814453125","226419033416748046875","49309033944091796875","8283917702607421875","1054316798513671875","98402901194609375","6358341307959375","254333652318375","4747561509943"],["23283064365386962890625","104308128356933593750000","219047069549560546875000","286221504211425781250000","260461568832397460937500","175030174255371093750000","89848822784423828125000","35939529113769531250000","11320951670837402343750","2817659082519531250000","552261180173828125000","84345343881093750000","9840290119460937500","847778841061250000","50866730463675000","1899024603977200","33232930569601"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832","1125899906842624","36028797018963968","1152921504606846976","36893488147419103232","1180591620717411303424","37778931862957161709568","1208925819614629174706176"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["625","350","49"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["25","7"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193967,"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-18ea3ef62fa5cfb6"]},{"id":193968,"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-18ea3ef62fa5cfb6","e-b4e581cbb1f9cfbb"]},{"id":193969,"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-18ea3ef62fa5cfb6","e-b4e581cbb1f9cfbb"]},{"id":193970,"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-18ea3ef62fa5cfb6"]},{"id":193971,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b4e581cbb1f9cfbb"]}],"region_exact":{},"incoming_edges":[{"id":"e-b4e581cbb1f9cfbb","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-1b09cdcaf124bad8","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"8aa62922d50f3ce53bac60c0479875464e4e312e37e1529187eb95a4e22876fc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-32cf9641286d8e9f","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","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]"}]},{"id":"e-18ea3ef62fa5cfb6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-1b09cdcaf124bad8","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":"709d6b6e2820084403e15ec72650a7df7d1dc1759781dd20b8b1673590c0fc04","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-32cf9641286d8e9f","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","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-c7fd36f954177034","kind":"fingerprint_related","node_a_id":"n-66be1d983d3de2fa","node_b_id":"n-1b09cdcaf124bad8","fingerprint":"f6-b7480fed0cfd4971","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b7480fed0cfd4971"},"other_id":"n-66be1d983d3de2fa","other":{"id":"n-66be1d983d3de2fa","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-b7480fed0cfd4971","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","25"],["49","350","625"]],"card_rows":[["1"],["7","25"],["49","350","625"],["343","3675","13125","15625"]],"row_sums_preview":["1","32","1024","32768","1048576","33554432","1073741824","34359738368"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["x \u21a6 x/(x+1)","x \u21a6 x/(x+1)"],"edges":["e-149dcf3d57e14e87","e-30c04d51b3c7d4af"],"text":"A038277 \u2192 x \u21a6 x/(x+1) \u2192 x \u21a6 x/(x+1)"},"headline":"A038277 \u2192 x \u21a6 x/(x+1) \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:              625,              350,                49;\n  1:            15625,            13125,              3675,              343;\n  2:           390625,           437500,            183750,            34300,             2401;\n  3:          9765625,         13671875,           7656250,          2143750,           300125,            16807;\n  4:        244140625,        410156250,         287109375,        107187500,         22509375,          2521050,          117649;\n  5:       6103515625,      11962890625,       10048828125,       4689453125,       1313046875,        220591875,        20588575,          823543;\n  6:     152587890625,     341796875000,      334960937500,     187578125000,      65652343750,      14706125000,      2058857500,       164708600,        5764801;\n  7:    3814697265625,    9613037109375,    10766601562500,    7034179687500,    2954355468750,     827219531250,    154414312500,     18529717500,     1297080225,      40353607;\n  8:   95367431640625,  267028808593750,   336456298828125,  251220703125000,  123098144531250,   41360976562500,   9650894531250,   1544143125000,   162135028125,   10088401750,   282475249;\n  9: 2384185791015625, 7343292236328125, 10280609130859375, 8635711669921875, 4835998535156250, 1895711425781250, 530799199218750, 106159839843750, 14862377578125, 1387155240625, 77680693475, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (25+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 350*x + 49*x^2.","mathematica":"P[0, x_] := 625 + 350*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[(25 + 7*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 625 + 350*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 7*x)*v[i - 1]); 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-7dcf11eddf2b9a8a","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-5a47242ff4b92549","e-18ea3ef62fa5cfb6"],"text":"A038321 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A038321 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}