{"id":"n-9ed77544d1e5ce22","revision":0,"first_index":0,"fingerprint":"f6-a7a31a2ddce7ca1a","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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":[["2025","1980","484"],["91125","133650","65340","10648"],["4100625","8019000","5880600","1916640","234256"],["184528125","451068750","441045000","215622000","52707600","5153632"],["8303765625","24357712500","29770537500","19405980000","7115526000","1391480640","113379904"],["373669453125","1278779906250","1875543862500","1528220925000","747130230000","219158200800","35714669760","2494357888"],["16815125390625","65765823750000","112532631750000","110031906600000","67241720700000","26298984096000","6428640556800","897968839680","54875873536"],["756680642578125","3329394827343750","6510816551250000","7427153695500000","5446579376700000","2662772139720000","867866475168000","181838690035200","22224728782080","1207269217792"],["34050628916015625","166469741367187500","366233431007812500","477459880425000000","408493453252500000","239649492574800000","97634978456400000","27275803505280000","5000563975968000","543271148006400","26559922791424"],["1532278301220703125","8240252197675781250","20142838705429687500","29542830101296875000","28886322765712500000","19771083137421000000","9665862867183600000","3375380683778400000","825093056034720000","134459609131584000","13147161781754880","584318301411328"],["68952523554931640625","404521471522265625000","1087713290093203125000","1772569806077812500000","1949826786685593750000","1525197842029620000000","869927658046524000000","364541113848067200000","111387562564687200000","24202729643685120000","3549733681073817600","315531882762117120","12855002631049216"],["3102863559971923828125","19720421736710449218750","57846570427683984375000","103695333655552031250000","126738741134563593750000","111530092198415962500000","72701097136745220000000","35542758600186552000000","13032344820068402400000","3539649210388948800000","692198067809394432000","92293075707919257600","7520176539163791360","282810057883082752"],["139628860198736572265625","955681976471352539062500","3036944947453409179687500","5938914563908889062500000","7984540691477506406250000","7807106453889117375000000","5725211399518686075000000","3198848274016789680000000","1368396206107182252000000","445995800509007548800000","109021195679979623040000","19381545898663044096000","2368855609836594278400","178170336466342133760","6221821273427820544"],["6283298708943145751953125","46077523865583068847656250","157687526117773168945312500","334063944219875009765625000","489960451522483347656250000","526979685637515422812500000","429390854963901455625000000","269902823120166629250000000","131952491303192574300000000","50174527557263349240000000","14717861416797249110400000","3270635870399388691200000","532992512213233712640000","60132488557390470144000","4199729359563778867200","136880068015412051968"],["282748441902441558837890625","2211721145547987304687500000","8109644200342620117187500000","18502003064485385156250000000","29397627091349000859375000000","34493215787182827675000000000","30916141557400904805000000000","21592225849613330340000000000","11875724217287331687000000000","5160808548747087350400000000","1766143370015669893248000000","470971565337511971532800000","95938652198382068275200000","14431797253773712834560000","1511902569442960392192000","98553648971096677416960","3011361496339065143296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["4489","300763","20151121","1350125107","90458382169","6060711605323","406067677556641","27206534396294947","1822837804551761449","122130132904968017083","8182718904632857144561","548242166610401428685587","36732225162896895721934329","2461059085914092013369600043","164890958756244164895763202881"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2025","1980","484"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["45","22"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194974,"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-103c94ffa29a7df8"]},{"id":194975,"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-103c94ffa29a7df8","e-a9bc275f22e7471c"]},{"id":194976,"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-103c94ffa29a7df8","e-a9bc275f22e7471c"]},{"id":194977,"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-103c94ffa29a7df8"]},{"id":194978,"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-a9bc275f22e7471c"]}],"region_exact":{},"incoming_edges":[{"id":"e-a9bc275f22e7471c","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9ed77544d1e5ce22","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":"96a87f70b78849c993361c8c33633b27549b69968f58be8d5252abb2ac1d4752","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-71831045520c8a2d","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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-103c94ffa29a7df8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9ed77544d1e5ce22","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":"436396b9f96c2db10a6e87c8a91046c01df59e48066bca68dfda626df29abb3b","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-71831045520c8a2d","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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":[],"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:                2025,                1980,                  484;\n  1:               91125,              133650,                65340,                10648;\n  2:             4100625,             8019000,              5880600,              1916640,               234256;\n  3:           184528125,           451068750,            441045000,            215622000,             52707600,              5153632;\n  4:          8303765625,         24357712500,          29770537500,          19405980000,           7115526000,           1391480640,           113379904;\n  5:        373669453125,       1278779906250,        1875543862500,        1528220925000,         747130230000,         219158200800,         35714669760,          2494357888;\n  6:      16815125390625,      65765823750000,      112532631750000,      110031906600000,       67241720700000,       26298984096000,       6428640556800,        897968839680,        54875873536;\n  7:     756680642578125,    3329394827343750,     6510816551250000,     7427153695500000,     5446579376700000,     2662772139720000,     867866475168000,     181838690035200,     22224728782080,      1207269217792;\n  8:   34050628916015625,  166469741367187500,   366233431007812500,   477459880425000000,   408493453252500000,   239649492574800000,   97634978456400000,   27275803505280000,   5000563975968000,    543271148006400,    26559922791424;\n  9: 1532278301220703125, 8240252197675781250, 20142838705429687500, 29542830101296875000, 28886322765712500000, 19771083137421000000, 9665862867183600000, 3375380683778400000, 825093056034720000, 134459609131584000, 13147161781754880, 584318301411328;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (45+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 2025 + 1980*x + 484*x^2.","mathematica":"P[0, x_] := 2025 + 1980*x + 484*x^2;\nP[n_, x_] := P[n, x] = Expand[(45 + 22*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] = 2025 + 1980*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (45 + 22*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-344e4bfa9933088a","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-9947ebaa7f991114","e-103c94ffa29a7df8"],"text":"A038337 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038337 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}
