{"id":"n-6e5997629441d544","revision":0,"first_index":0,"fingerprint":"f6-e543af4f498eab8b","name":"theta_plus_one_squared 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":false,"rows":[["144","768","576"],["1728","13824","20736","8192"],["20736","221184","497664","393216","102400"],["248832","3317760","9953280","11796480","6144000","1179648"],["2985984","47775744","179159040","283115520","221184000","84934656","12845056"],["35831808","668860416","3009871872","5945425920","6193152000","3567255552","1078984704","134217728"],["429981696","9172942848","48157949952","114152177664","148635648000","114152177664","51791265792","12884901888","1358954496"],["5159780352","123834728448","743008370688","2054739197952","3210529996800","3082108796928","1864485568512","695784701952","146767085568","13421772800"],["61917364224","1651129712640","11145125560320","35224100536320","64210599936000","73970611126272","55934567055360","27831388078080","8806025134080","1610612736000","129922760704"],["743008370688","21794912206848","163461841551360","581197658849280","1210828455936000","1627353444777984","1476672570261504","918435806576640","387465105899520","106300440576000","17149804412928","1236950581248"],["8916100448256","285315214344192","2353850518339584","9299162541588480","21794912206848000","33476985149718528","35440141686276096","26450951229407232","13948743812382720","5102421147648000","1234785917730816","178120883699712","11613591568384"],["106993205379072","3709097786474496","33381880078270464","145066935648780288","377778478252032000","652801210419511296","789808871865581568","687724731964588032","435200806946340864","198994424758272000","64208867722002432","13893428928577536","1811720284667904","107752139522048"],["1283918464548864","47932956009824256","467346321095786496","2215567744454098944","6346678434634137600","12185622594497544192","16585986309177212928","16505393567150112768","12185622594497544192","6686212671877939200","2696772444324102144","778032020000342016","152184503912103936","18102359439704064","989560464998400"],["15407021574586368","616280862983454720","6470949061326274560","33233516166811484160","103854738021285888000","219341206700955795456","331719726183544258560","371371355260877537280","313344581001365422080","200586380156338176000","97083807995667677184","35011440900015390720","9131070234726236160","1629212349573365760","178120883699712000","9007199254740992"],["184884258895036416","7888395046188220416","88744444269617479680","490833469540600381440","1661675808340574208000","3828501062416682975232","6369018742724049764352","7922588912232054128640","7520269944032770129920","5501797855716704256000","3106681855861365669888","1344439330560591003648","438291371266859335680","104269590372695408640","17099604835172352000","1729382256910270464","81346268269379584"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1488","44480","1235200","32640000","832000000","20633600000","500736000000","11939840000000","280576000000000","6512640000000000","149585920000000000","3404595200000000000","76873728000000000000","1723596800000000000000","38404096000000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","768","576"]],"recurrence":{"denominator":{"coeffs":[["9","30","16"],["0","18","16"],["0","0","4"]]},"terms":[{"coeff":{"coeffs":[["108","648","816","288"],["0","216","432","192"],["0","0","48","32"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193274,"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-7ccea14f95b0cd14"]},{"id":193275,"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-7ccea14f95b0cd14"]},{"id":193276,"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-7ccea14f95b0cd14"]},{"id":193277,"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-7ccea14f95b0cd14"]}],"region_exact":{},"incoming_edges":[{"id":"e-7ccea14f95b0cd14","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-6e5997629441d544","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"fa13707623b8d0c367e14066ac50f478f8f4842171e201aba85c227dc13b6625","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-6874f381e9c7e0a2","role":"sequence","name":"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-23aeb6031914e957","kind":"fingerprint_related","node_a_id":"n-ad2b43f723b42ee4","node_b_id":"n-6e5997629441d544","fingerprint":"f6-e543af4f498eab8b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e543af4f498eab8b"},"other_id":"n-ad2b43f723b42ee4","other":{"id":"n-ad2b43f723b42ee4","name":"theta_plus_one_squared of basis_map_A013611 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-e543af4f498eab8b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","16"],["36","192","144"]],"card_rows":[["1"],["6","16"],["36","192","144"],["216","1728","2592","1024"]],"row_sums_preview":["1","22","372","5560","77200","1020000","13000000","161200000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-0e13ca82ccc1a33c","e-6fa3983c3b07eab2"],"text":"A038243 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038243 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-6504f884fe0ca37f","kind":"fingerprint_related","node_a_id":"n-cdb87b58d20a5654","node_b_id":"n-6e5997629441d544","fingerprint":"f6-e543af4f498eab8b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e543af4f498eab8b"},"other_id":"n-cdb87b58d20a5654","other":{"id":"n-cdb87b58d20a5654","name":"theta_plus_one_squared of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-e543af4f498eab8b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["9","24"],["27","144","108"]],"card_rows":[["3"],["9","24"],["27","144","108"],["81","648","972","384"]],"row_sums_preview":["3","33","279","2085","14475","95625","609375","3778125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-618e7f7547136f14","e-ebb256587128c62c"],"text":"A120910 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A120910 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-c3958787ef820f3d","kind":"fingerprint_related","node_a_id":"n-0ee65094b134f8b2","node_b_id":"n-6e5997629441d544","fingerprint":"f6-e543af4f498eab8b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e543af4f498eab8b"},"other_id":"n-0ee65094b134f8b2","other":{"id":"n-0ee65094b134f8b2","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-e543af4f498eab8b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","8"],["9","48","36"]],"card_rows":[["1"],["3","8"],["9","48","36"],["27","216","324","128"]],"row_sums_preview":["1","11","93","695","4825","31875","203125","1259375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["(\u03b8+1)\u00b2"],"edges":["e-635f90021d26fb2c"],"text":"A038220 \u2192 (\u03b8+1)\u00b2"},"headline":"A038220 \u2192 (\u03b8+1)\u00b2","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:          144,            768,             576;\n  1:         1728,          13824,           20736,            8192;\n  2:        20736,         221184,          497664,          393216,           102400;\n  3:       248832,        3317760,         9953280,        11796480,          6144000,          1179648;\n  4:      2985984,       47775744,       179159040,       283115520,        221184000,         84934656,         12845056;\n  5:     35831808,      668860416,      3009871872,      5945425920,       6193152000,       3567255552,       1078984704,       134217728;\n  6:    429981696,     9172942848,     48157949952,    114152177664,     148635648000,     114152177664,      51791265792,     12884901888,      1358954496;\n  7:   5159780352,   123834728448,    743008370688,   2054739197952,    3210529996800,    3082108796928,    1864485568512,    695784701952,    146767085568,     13421772800;\n  8:  61917364224,  1651129712640,  11145125560320,  35224100536320,   64210599936000,   73970611126272,   55934567055360,  27831388078080,   8806025134080,   1610612736000,   129922760704;\n  9: 743008370688, 21794912206848, 163461841551360, 581197658849280, 1210828455936000, 1627353444777984, 1476672570261504, 918435806576640, 387465105899520, 106300440576000, 17149804412928, 1236950581248;\n","recurrence_sentence":"The row polynomials satisfy (9+(18*n+30)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = (108+(216*n+648)*x+(48*n^2+432*n+816)*x^2+(32*n^2+192*n+288)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 768*x + 576*x^2.","mathematica":"P[0, x_] := 144 + 768*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((108 + (216*n + 648)*x + (48*n^2 + 432*n + 816)*x^2 + (32*n^2 + 192*n + 288)*x^3)*P[n - 1, x])/(9 + (18*n + 30)*x + (4*n^2 + 16*n + 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 768*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((108 + (216*n + 648)*x + (48*n^2 + 432*n + 816)*x^2 + (32*n^2 + 192*n + 288)*x^3)*v[i - 1])/(9 + (18*n + 30)*x + (4*n^2 + 16*n + 16)*x^2)); 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":["(\u03b8+1)\u00b2"],"edges":["e-7ccea14f95b0cd14"],"text":"A038334 \u2192 (\u03b8+1)\u00b2"},"headline":"A038334 \u2192 (\u03b8+1)\u00b2","chain":null}