{"id":"n-3d8484980ed9ab61","revision":0,"first_index":0,"fingerprint":"f6-9c4ccaa67290bb6b","name":"shift_x_plus_one of Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + 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"],["7","7","1"],["30","45","17","1"],["157","314","194","37","1"],["972","2430","2094","711","77","1"],["6961","20883","23128","11451","2402","157","1"],["56660","198310","268590","175700","56200","7755","317","1"],["516901","2067604","3312311","2700319","1173353","258379","24374","637","1"],["5225670","23515515","43506220","42532700","23473932","7222615","1139084","75351","1277","1"],["57999271","289996355","608575530","694323990","465920096","185611014","42091248","4882751","230522","2557","1"],["701216922","3856693071","9052740585","11812134600","9347959320","4591414317","1374575271","236084430","20529720","700515","5117","1"],["9173819257","55042915542","142902342101","209951651370","191601089366","112166520206","41989629378","9710932205","1288184023","85154019","2119454","10237","1"],["129134686520","839375462380","2388252525372","3902258803366","4036687501280","2747847859785","1239215186260","363514590782","66251311518","6887920445","349799524","6394191","20477","1"],["1946194117057","13623358819399","42157109098419","75838989938327","87729962091842","68149121220222","36037723563989","12873538777919","3019360432328","440208729820","36275873318","1426904551","19254242","40957","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","93","703","6285","64983","763533","10053879","146692365","2349633335","40994053869","773914367159","15719782061901","341853288862391"],"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":[["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178853,"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-5c87df9cdc528431"]},{"id":178854,"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-5c87df9cdc528431"]},{"id":178855,"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-5c87df9cdc528431"]},{"id":178856,"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-5c87df9cdc528431"]}],"region_exact":{},"incoming_edges":[{"id":"e-5c87df9cdc528431","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3d8484980ed9ab61","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":"23edeb9c3e691787927403d85c93d3aab608cfaacc199f4b173d8d4f3facc677","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-2872b2f2cd7c5725","role":"sequence","name":"Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + 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-dcd93e2dee6458c1","kind":"fingerprint_related","node_a_id":"n-3d8484980ed9ab61","node_b_id":"n-f63b8f7fcf81ceba","fingerprint":"f6-9c4ccaa67290bb6b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c4ccaa67290bb6b"},"other_id":"n-f63b8f7fcf81ceba","other":{"id":"n-f63b8f7fcf81ceba","name":"mobius_x_over_x_plus_one of Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","first_index":0,"fingerprint":"f6-9c4ccaa67290bb6b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","7","7"]],"card_rows":[["1"],["1","2"],["1","7","7"],["1","17","45","30"]],"row_sums_preview":["1","3","15","93","703","6285","64983","763533"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2872b2f2cd7c5725","steps":["x \u21a6 x/(x+1)"],"edges":["e-c6bed485de2bb185"],"text":"A144438 \u2192 x \u21a6 x/(x+1)"},"headline":"A144438 \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:       7,        7,        1;\n  3:      30,       45,       17,        1;\n  4:     157,      314,      194,       37,        1;\n  5:     972,     2430,     2094,      711,       77,       1;\n  6:    6961,    20883,    23128,    11451,     2402,     157,       1;\n  7:   56660,   198310,   268590,   175700,    56200,    7755,     317,     1;\n  8:  516901,  2067604,  3312311,  2700319,  1173353,  258379,   24374,   637,    1;\n  9: 5225670, 23515515, 43506220, 42532700, 23473932, 7222615, 1139084, 75351, 1277, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + (n+1+n*x)*P(n-1,x) - (x+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[(1 + x)*P[n - 2, x] + (n + 1 + n*x)*P[n - 1, x] - (x + 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] = (1 + x)*v[i - 2] + (n + 1 + n*x)*v[i - 1] - (x + 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-2872b2f2cd7c5725","steps":["x \u21a6 x+1"],"edges":["e-5c87df9cdc528431"],"text":"A144438 \u2192 x \u21a6 x+1"},"headline":"A144438 \u2192 x \u21a6 x+1","chain":null}
