{"id":"n-80b2fb0627fa560a","revision":0,"first_index":0,"fingerprint":"f6-374a4446546c6a5d","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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":[["1040","1152","256"],["27200","49920","27648","4096"],["672000","1740800","1597440","589824","65536"],["16000000","53760000","69632000","42598400","11796480","1048576"],["371200000","1536000000","2580480000","2228224000","1022361600","226492416","16777216"],["8448000000","41574400000","86016000000","96337920000","62390272000","22900899840","4227858432","268435456"],["189440000000","1081344000000","2660761600000","3670016000000","3082813440000","1597190963200","488552529920","77309411328","4294967296"],["4198400000000","27279360000000","77856768000000","127716556800000","132120576000000","88785027072000","38332583116800","10050223472640","1391569403904","68719476736"],["92160000000000","671744000000000","2182348800000000","4152360960000000","5108662272000000","4227858432000000","2367600721920000","876173328384000","201004469452800","24739011624960","1099511627776"],["2007040000000000","16220160000000000","59113472000000000","128031129600000000","182703882240000000","179824911974400000","124017180672000000","59528246722560000","19275813224448000","3930754069299200","435406604599296","17592186044416"],["43417600000000000","385351680000000000","1557135360000000000","3783262208000000000","6145494220800000000","7015829078016000000","5754397183180800000","3401614098432000000","1428677921341440000","411217348788224000","75470478130544640","7599824371187712","281474976710656"],["933888000000000000","9030860800000000000","40076574720000000000","107961384960000000000","196729634816000000000","255652559585280000000","243215408037888000000","170987802014515200000","88441966559232000000","33018334182113280000","8553320854795059200","1427078131923025920","131730289100587008","4503599627370496"],["19988480000000000000","209190912000000000000","1011456409600000000000","2992384245760000000000","6045837557760000000000","8813487639756800000000","9544362224517120000000","7782893057212416000000","4787658456406425600000","2201222278807552000000","739610685679337472000","174176715588553932800","26638791795896483840","2269814212194729984","72057594037927936"],["425984000000000000000","4797235200000000000000","25102909440000000000000","80916512768000000000000","179543054745600000000000","290200202772480000000000","352539505590272000000000","327235276269158400000000","233486791716372480000000","127670892170838016000000","52829334691381248000000","16136960414821908480000","3483534311771078656000","491793079308858163200","38911100780481085440","1152921504606846976"],["9043968000000000000000","109051904000000000000000","614046105600000000000000","2142114938880000000000000","5178656817152000000000000","9192604402974720000000000","12381875318292480000000000","12892873347301376000000000","10471528840613068800000000","6641402075487928320000000","3268374839573453209600000","1229482698272145408000000","344255155516200714240000","68598829524107395072000","8992787735933406412800","664082786653543858176","18446744073709551616"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2448","108864","4665600","194835456","7981535232","322163785728","12851722911744","507799783342080","19905751507009536","775105589293350912","30009748476036907008","1156165046550474522624","44351249326690334146560","1694900051192289077231616","64553584558454140506734592"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1040","1152","256"]],"recurrence":{"denominator":{"coeffs":[["9","4"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["260","288","64"],["80","64","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194683,"position":0,"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-95fc073445189c71"]},{"id":194684,"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-95fc073445189c71"]},{"id":194685,"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-95fc073445189c71"]},{"id":194686,"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-95fc073445189c71"]}],"region_exact":{},"incoming_edges":[{"id":"e-95fc073445189c71","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-80b2fb0627fa560a","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"3c15d0eaaec6043f6c983adfedbc6fdb942f2b25f6965e392ad817ea05f46a15","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-7e007679d5feba96","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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-7cd9c281bff5668b","kind":"fingerprint_related","node_a_id":"n-5fb9511402d2ef0d","node_b_id":"n-80b2fb0627fa560a","fingerprint":"f6-374a4446546c6a5d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-374a4446546c6a5d"},"other_id":"n-5fb9511402d2ef0d","other":{"id":"n-5fb9511402d2ef0d","name":"one_plus_derivative of basis_map_A007318 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-374a4446546c6a5d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","4"],["65","72","16"]],"card_rows":[["1"],["9","4"],["65","72","16"],["425","780","432","64"]],"row_sums_preview":["1","13","153","1701","18225","190269","1948617","19663317"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["basis A007318","D+1"],"edges":["e-13c78408b476aecd","e-3296652b0fbb530a"],"text":"A013611 \u2192 basis A007318 \u2192 D+1"},"headline":"A013611 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-933591e07fb45049","kind":"fingerprint_related","node_a_id":"n-1f4836f47af1fcb8","node_b_id":"n-80b2fb0627fa560a","fingerprint":"f6-374a4446546c6a5d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-374a4446546c6a5d"},"other_id":"n-1f4836f47af1fcb8","other":{"id":"n-1f4836f47af1fcb8","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-374a4446546c6a5d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["585","648","144"],["11475","21060","11664","1728"],["212625","550800","505440","186624","20736"]],"card_rows":[["585","648","144"],["11475","21060","11664","1728"],["212625","550800","505440","186624","20736"],["3796875","12757500","16524000","10108800","2799360","248832"]],"row_sums_preview":["1377","45927","1476225","46235367","1420541793","43003674279","1286623443969","38127987424935"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013611","D+1"],"edges":["e-a5d7492b5780a0c3","e-b89ebfd5012cb92b"],"text":"A038329 \u2192 basis A013611 \u2192 D+1"},"headline":"A038329 \u2192 basis A013611 \u2192 D+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:             1040,              1152,               256;\n  1:            27200,             49920,             27648,               4096;\n  2:           672000,           1740800,           1597440,             589824,              65536;\n  3:         16000000,          53760000,          69632000,           42598400,           11796480,            1048576;\n  4:        371200000,        1536000000,        2580480000,         2228224000,         1022361600,          226492416,           16777216;\n  5:       8448000000,       41574400000,       86016000000,        96337920000,        62390272000,        22900899840,         4227858432,         268435456;\n  6:     189440000000,     1081344000000,     2660761600000,      3670016000000,      3082813440000,      1597190963200,       488552529920,       77309411328,        4294967296;\n  7:    4198400000000,    27279360000000,    77856768000000,    127716556800000,    132120576000000,     88785027072000,     38332583116800,    10050223472640,     1391569403904,      68719476736;\n  8:   92160000000000,   671744000000000,  2182348800000000,   4152360960000000,   5108662272000000,   4227858432000000,   2367600721920000,   876173328384000,   201004469452800,   24739011624960,   1099511627776;\n  9: 2007040000000000, 16220160000000000, 59113472000000000, 128031129600000000, 182703882240000000, 179824911974400000, 124017180672000000, 59528246722560000, 19275813224448000, 3930754069299200, 435406604599296, 17592186044416;\n","recurrence_sentence":"The row polynomials satisfy (4*n+9+4*x)*P(n,x) = (80*n+260+(64*n+288)*x+64*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1040 + 1152*x + 256*x^2.","mathematica":"P[0, x_] := 1040 + 1152*x + 256*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((80*n + 260 + (64*n + 288)*x + 64*x^2)*P[n - 1, x])/(4*n + 9 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1040 + 1152*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 260 + (64*n + 288)*x + 64*x^2)*v[i - 1])/(4*n + 9 + 4*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-6874f381e9c7e0a2","steps":["basis A013609","D+1"],"edges":["e-4eca5e8e987e069c","e-95fc073445189c71"],"text":"A038334 \u2192 basis A013609 \u2192 D+1"},"headline":"A038334 \u2192 basis A013609 \u2192 D+1","chain":null}
