{"id":"n-c543221da2eb63df","revision":0,"first_index":0,"fingerprint":"f6-4285d334207d89bf","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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"],["6","4"],["35","48","16"],["205","424","288","64"],["1200","3320","3408","1536","256"],["7025","24340","33456","22784","7680","1024"],["41125","171200","294960","269056","136960","36864","4096"],["240750","1170300","2424240","2772864","1890560","768000","172032","16384"],["1409375","7835000","18962000","26086400","22305280","12134400","4100096","786432","65536"],["8250625","51627000","142952400","229841280","236416256","161295360","72970240","21102592","3538944","262144"],["48300000","335950000","1047420000","1927296000","2317318400","1902034944","1079070720","417726464","105578496","15728640","1048576"],["282753125","2164087500","7502170000","15547176000","21397222400","20531590144","14013526016","6802309120","2300903424","516423680","69206016","4194304"],["1655265625","13824400000","52748100000","121570720000","188470592000","207026288640","165197484032","96470564864","40912551936","12286689280","2479882240","301989888","16777216"],["9690093750","87694512500","365205300000","926590560000","1597810656000","1977260651520","1806286565376","1233208410112","629158772736","236869386240","63957893120","11727273984","1308622848","67108864"],["56726796875","552976000000","2495842650000","6912620000000","13124051360000","18066559692800","18594568007680","14534331072512","8669072523264","3926065676288","1328813178880","325897420800","54744055808","5637144576","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","99","981","9720","96309","954261","9455130","93684519","928256841","9197472240","91131561729","902961305721","8946835807050","88648174014939"],"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"]],"recurrence":{"denominator":{"coeffs":[["4"],["4"]]},"terms":[{"coeff":{"coeffs":[["56","32"],["-8","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["45","56","16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-7bfc29c470808344","recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179755,"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-959a0b874aaa1477"]},{"id":179756,"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-4322e148050fbf70","e-959a0b874aaa1477"]},{"id":179757,"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-4322e148050fbf70","e-959a0b874aaa1477"]},{"id":179758,"position":3,"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-4322e148050fbf70"]},{"id":179759,"position":4,"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-4322e148050fbf70"]}],"region_exact":{},"incoming_edges":[{"id":"e-4322e148050fbf70","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c543221da2eb63df","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":"c5ccf9b16921c76a516e10779ca3425a7a01f1df736cef54c3109b68bd849a07","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0552ada0abe44307","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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-7bfc29c470808344","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-c543221da2eb63df","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"bdf12ee35e27b340edb8a5deb0b2ec9adbee91dc16c5d646675af07585572275","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-959a0b874aaa1477","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c543221da2eb63df","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":"2614c0014fa687ea76d7ffff302bfca264038e619db288fcdc64b901bc498c78","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-0552ada0abe44307","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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:       1;\n  1:       6,        4;\n  2:      35,       48,        16;\n  3:     205,      424,       288,        64;\n  4:    1200,     3320,      3408,      1536,       256;\n  5:    7025,    24340,     33456,     22784,      7680,      1024;\n  6:   41125,   171200,    294960,    269056,    136960,     36864,     4096;\n  7:  240750,  1170300,   2424240,   2772864,   1890560,    768000,   172032,    16384;\n  8: 1409375,  7835000,  18962000,  26086400,  22305280,  12134400,  4100096,   786432,   65536;\n  9: 8250625, 51627000, 142952400, 229841280, 236416256, 161295360, 72970240, 21102592, 3538944, 262144;\n","recurrence_sentence":"The row polynomials satisfy (4*n+4)*P(n,x) = -(8*n-56-32*x)*P(n-1,x) + (45+56*x+16*x^2)*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[(-(8*n - 56 - 32*x)*P[n - 1, x] + (45 + 56*x + 16*x^2)*D[P[n - 1, x], x])/(4*n + 4)]];\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] = (-(8*n - 56 - 32*x)*v[i - 1] + (45 + 56*x + 16*x^2)*deriv(v[i - 1], x))/(4*n + 4)); 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-21854f4257effb01","steps":["basis A013611","basis A007318"],"edges":["e-3723cb941a96ee35","e-4322e148050fbf70"],"text":"A030528 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A030528 \u2192 basis A013611 \u2192 basis A007318","chain":null}